On the Fundamental Impossibility of Hallucination Control in Large Language Models

This paper establishes a fundamental impossibility theorem: no LLM capable of performing non-trivial knowledge aggregation can simultaneously achieve truthful knowledge representation, semantic information conservation, complete revelation of relevant knowledge, and knowledge-constrained optimality. The impossibility is not an engineering limitation but arises from the mathematical structure of information aggregation itself. We establish this result by describing the inference process as an auction of ideas, where distributed components compete exploiting their partial knowledge to shape responses. The proof spans three independent mathematical domains: mechanism design theory (Green-Laffont), the theory of proper scoring rules (Savage), and direct architectural analysis of transformers (Log-Sum-Exp convexity). In particular, we show how to quantify the creation of overconfident or intuitive responses-the signature of both hallucination and creativity, or imagination. To support this analysis, we introduce the complementary concepts of the semantic information measure and the emergence operator to model bounded reasoning in a general setting. We prove that while bounded reasoning generates accessible information, providing valuable insights and inspirations, the idealized unconstrained reasoning strictly preserves semantic content. By demonstrating that hallucination and imagination are mathematically identical phenomena-grounded in departures from truthfulness, semantic information conservation, revelation of relevant knowledge, and knowledge-constrained optimality-we offer a principled foundation for managing these behaviors in advanced AI systems. Finally, we present some speculative ideas to inspire evaluation and refinements of the proposed theory.