Singularity Theory in the Output Space of Generative AI: On Global Continuity and Local Optimality under Prompt Variation

Abstract: This paper establishes a theoretical framework for understanding the behavior of Large Language Models (LLMs) through the lens of Singularity Theory and Information Geometry. Moving beyond the "black box" view of generative AI, the author treats the output space of AI—shaped by continuous prompt variations—as a manifold where local optimality and global continuity intersect. Drawing analogies from the history of thermodynamics and the work of Shun-ichi Amari on information geometry, this study explores how "phase transitions" in AI responses can be understood as singularities. By integrating concepts from optimal transport (Monge-Kantorovich) and variational analysis, the paper aims to provide a principled basis for AI reliability and the "laws" governing the latent spaces of generative models. This work is a product of an innovative collaboration between an independent researcher and multiple AI entities.