Sheaf-Theoretic Semantics and Quantum Contextuality in Large Language Models: A Unified Categorical Framework for Understanding Semantic Coherence and Hallucination Phenomena

Abstract

This paper introduces a novel theoretical framework for understanding semantic processing in Large Language Models through the lens of sheaf theory and quantum contextuality. We propose that the semantic structure of LLM-generated text can be rigorously modeled as a sheaf over the topological space of prompt contexts, where local semantic sections may fail to glue into globally consistent interpretations. This mathematical formalization reveals a deep structural parallel between LLM behavior and quantum mechanical systems exhibiting contextuality, as characterized by the Kochen-Specker theorem. Our central thesis posits that hallucination phenomena in LLMs arise fundamentally from cohomological obstructions to the existence of global semantic sections, analogous to how quantum systems exhibit measurement outcomes that cannot be explained by preexisting hidden variables. We develop a complete categorical framework using topos-theoretic methods, introduce formal definitions of semantic presheaves and their associated Grothendieck topologies, and demonstrate how Cech cohomology groups can quantify the degree of semantic inconsistency in model outputs. This work bridges abstract mathematics with practical AI interpretability, offering new diagnostic tools for understanding when and why language models produce incoherent or factually inconsistent outputs.