Language Model Hallucinations: An Impossibility Theorem and Its Architectural Consequences

We establish a structural account of hallucination in autoregressive language models: no system that selects tokens solely by bounded-context next-token prediction can guarantee globally consistent outputs across all prompts whose correctness depends on constraints extending beyond the context window. We formalize language generation as a sequential decision process under a terminal consistency constraint and prove that a structural failure class—which we call delayed constraint failure—necessarily arises in any bounded-context generator: locally plausible token choices can irreversibly eliminate all globally valid completions while remaining indistinguishable within the model's evaluation window. From this result we derive a necessary condition for reliable generation—the constraint requirement—stating that any system achieving bounded inconsistency must incorporate a mechanism that excludes token continuations lacking any valid completion. Controlled empirical evidence for the predicted behaviour—a monotonically rising failure-time distribution concentrated near the terminal step, distinguishing structural hallucination from random token error—is available in the companion framework's Appendix A. Building on this, we classify common language-model mitigation strategies—retrieval augmentation, chain-of-thought prompting, grammar-constrained decoding, verifier-reranking, and process supervision—by the structural mechanism through which each acts on the forward-local obstruction, identifying which satisfy the constraint requirement and on which constraint sub-class. The result identifies a structural failure mode that contributes to hallucination independently of training quality or model scale. We do not claim that delayed constraint failure is the sole or dominant cause of hallucination in modern language models; rather, it constitutes a necessary and unavoidable failure class under bounded-context generation, and solutions addressing it must operate at the level of global feasibility rather than local likelihood.