Toward a Mathematical Axiom System for Meaning Spaces: Information Geometry, Sheaf Theory, and the Open Truth Hypothesis

Large language models lack a rigorous mathematical framework for the concept of"meaning" in their representational spaces. This position paper argues that such aframework can be constructed by synthesizing two mathematical disciplines:information geometry and sheaf theory. We survey the proven capabilities ofinformation geometry — the Fisher-Rao metric, dual connections, natural gradientdescent, and applications to LLM engineering — and identify its structural limitations:singularity of neural network parameter spaces, computational intractability at scale,inherent locality, and the fundamental gap between statistical distinguishability andsemantic meaning. We propose that sheaf theory provides the missing local-to-globalorganizational structure, enabling local statistical geometries to be assembled intoglobal semantic structures via gluing conditions, with sheaf cohomology formalizingphenomena such as hallucination and contextual inconsistency. To our knowledge, noexisting work combines these two frameworks. We present four preliminary axioms for"open meaning spaces" grounded in the philosophical commitment that meaning, liketruth, is an ever-expanding creative process rather than a closed totality, and providean honest roadmap of open problems. This paper is an invitation to interdisciplinarycollaboration among information geometers, algebraic topologists, and LLMresearchers.