This paper proves that perfect hallucination control is mathematically impossible in large-scale language models (LLMs). No LLM inference mechanism can simultaneously achieve truthful response generation, semantic information preservation, relevant knowledge disclosure, and knowledge constraint optimization. This impossibility is not an engineering limitation, but a fundamental problem that arises from the mathematical structure of information aggregation itself. Using three mathematical frameworks—auction theory, appropriate score theory for probabilistic prediction, and log-sum exponential analysis for Transformer architectures—we show that information aggregation inevitably violates the preservation principle. The Jensen gap of Transformer probability aggregation is a direct measure of this impossibility. These results redefine hallucination as an inevitable mathematical feature of distributed intelligence, not an engineering error. There is a fundamental tradeoff between truthfulness, knowledge utilization, and response completeness, and they provide a principled foundation for managing hallucinations rather than eliminating them. This study reveals deep connections between classical results in neural network inference, the philosophy of knowledge and inference, game theory, and information theory, and suggests new research directions for developing beneficial AI systems within mathematical constraints.
