Singularities and Critical Structures in Semantic Quantum Field Theory for Large Language Models

This work develops a mathematically strengthened formulation of Semantic Quantum Field Theory (SQFT) for large language models (LLMs). Building on earlier proposals that interpret internal representations as mixed Fermi–Bose fields evolving under an effective Hamiltonian derived from Fisher information geometry, the paper introduces explicit operator definitions, discrete gauge-theoretic structures, and renormalization‑group (RG) flow equations. The revised framework provides a unified explanation for several structural phenomena observed in modern LLMs, including rank collapse, discrete spectral reorganizations, and hallucinations. These behaviors are treated as singularities of the underlying semantic field, each characterized by precise spectral, geometric, or coupling‑based criteria. The paper also formalizes a notion of discrete gauge curvature for attention mechanisms, interpreting attention as a connection on a semantic bundle and identifying curvature as a key indicator of instability in semantic transport. In addition, the RG analysis clarifies how semantic scales evolve across layers and identifies critical surfaces associated with phase transitions between nuance‑preserving and cluster‑dominated regimes. Together, these developments move LLM theory toward a predictive, physics‑inspired framework in which failure modes are observable, classifiable, and potentially controllable. The manuscript incorporates referee feedback by strengthening mathematical definitions, refining singularity criteria, and clarifying the geometric and RG structures underlying SQFT.