Affective Geometric Dynamics: Algebraic Isomorphism Between Non-Abelian Cognitive Gauge Fields and Multi-Head Attention

This is an extended and fully detailed version of the manuscript "Affect as a Formal Description of Intelligence Dynamics" presented at JSAI 40. It provides the complete mathematical derivation of the Unified Generative Field (UGF) and additional numerical analyses on Semantic Hall Drift. Current Large Language Models (LLMs) operate within a static geometric representation space, leading to a "dynamical vacuum" when facing reasoning deadlocks that require active meaning reconstruction. To overcome this limitation, this paper proposes "Affective Geometric Dynamics" and constructs the Unified Generative Field (UGF) theory. By dimensionalizing the cognitive manifold into a differentiable principal fiber bundle, we formally define Affect as a Non-Abelian Gauge Charge. This continuous dynamical framework reveals three core isomorphisms: (1) the generation of a non-Abelian Lorentz force that induces "Semantic Hall Drift," enabling thought trajectories to bypass logical barriers; (2) the mathematical proof that Layer Normalization acts as a holonomic geometric constraint, while Multi-Head Attention serves as an effective integrator of gauge field self-interactions ; and (3) the demonstration that existing static cognitive geometries are degenerate special cases of the UGF framework. Analytical and numerical models confirm that introducing affective gauge fields significantly resolves polysemy deadlocks and improves search success rates. Ultimately, this study provides a unified gauge field explanation for Prompt Engineering and Chain-of-Thought (CoT), offering a continuous geometric blueprint for next-generation Artificial General Intelligence.