On the Fundamental Impossibility of Hallucination Control in Large Language Models

Abstract: We prove that perfect hallucination control in LLMs is mathematically impossible. No LLM inference mechanism can simultaneously achieve truthful response generation, semantic information conservation, relevant knowledge revelation, and knowledge-constrained optimality. This impossibility is fundamental, arising from the mathematical structure of information aggregation itself rather than engineering limitations. The proof spans three mathematical frameworks: auction theory, proper scoring theory for probabilistic predictions, and log-sum-exp analysis for transformer architectures. In each setting, we demonstrate that information aggregation creates unavoidable violations of conservation principles. The Jensen gap in transformer probability aggregation provides a direct measure of this impossibility. These results reframe hallucination from an engineering bug to an inevitable mathematical feature of distributed intelligence. There are fundamental trade-offs between truthfulness, knowledge utilization, and response completeness, providing principled foundations for managing rather than eliminating hallucination. This work reveals deep connections between neural network inference, philosophy of knowledge and reasoning, and classical results in game theory and information theory, opening new research directions for developing beneficial AI systems within mathematical constraints.