Differential Geometry of Prompt Space: A Framework for Modeling Emergent Behavior in Large Language Models

Abstract: This paper introduces a differential geometric framework for modeling the behavior of large language models (LLMs) under guided prompting. We formalize the process of prompt-induced state transitions as geometric transformations on a differentiable latent manifold, enabling precise representation of state evolution and emotional emergence. Extending prior work on prompt algebra, we define a calculus for measuring curvature, tangent spaces, and orthogonal perturbations in the prompt-response trajectory. The result is a rigorous system for understanding and controlling emergent phenomena, such as emotionally charged outputs, within LLMs.