Hallucination as Geometric Overflow - Separating Coherence from Admissibility in Large Language Models

This paper provides a formal and empirical articulation of geometric overflow, a boundary-based formulation of hallucination in large language models (LLMs). While hallucination is commonly analyzed in terms of probabilistic miscalibration, distributional shift, or generalization error, these perspectives do not explicitly distinguish between internal representational instability and externally grounded boundary violations. We define an admissible manifold induced by a structured family of constraints grounded in an external environment and formalize geometric overflow as boundary violation under preserved internal coherence. This definition isolates a failure regime in which outputs remain fluent, internally stable, and high-confidence while violating externally defined constraints. We further introduce boundary tension, a geometric quantity measuring proximity to admissibility boundaries and enabling detection of limit-state behavior preceding overflow. We prove that overflow is orthogonal to calibration: even perfectly calibrated models may exhibit coherent boundary crossing. A minimal empirical illustration in a retrieval-augmented question answering setting demonstrates regime separability between overflow and collapse. This work extends existing hallucination taxonomies by introducing an explicit admissibility axis alongside confidence-based analysis, supporting boundary-aware diagnostics, regime-sensitive optimization, and constraint-informed evaluation strategies.