Hallucination Is a Mathematical Inevitability

Hallucination in autoregressive language models is a proven mathematical limitation, not an engineering problem awaiting a fix. Xu et al. (2024) use computability theory to show that LLMs cannot learn all computable functions, making hallucination inevitable regardless of scale, training data, or architecture. This is not a “we haven’t solved it yet” result. It is a formal impossibility proof: no autoregressive model can guarantee freedom from hallucination.

This matters for guardrail erosion because hallucination is one of the four suggestible actor properties. If it cannot be eliminated, the erosion dynamics that follow from it cannot be eliminated either. A more capable model hallucinates differently, not less.