On the Fundamental Impossibility of Hallucination Control in Large Language Models

• URL: http://arxiv.org/abs/2506.06382v4
• Date: Wed, 06 Aug 2025 11:34:54 GMT
• Title: On the Fundamental Impossibility of Hallucination Control in Large Language Models
• Authors: Michał P. Karpowicz,
• Abstract summary: This paper establishes a fundamental impossibility theorem: no LLM capable performing non-trivial knowledge aggregation can simultaneously achieve truthful (internally consistent) knowledge representation.<n>This impossibility is not an engineering limitation but arises from the mathematical structure of information aggregation itself.<n>By demonstrating that hallucination and imagination are mathematically identical phenomena-grounded in the necessary violation of information conservation, this paper offers a principled foundation for managing these behaviors in advanced AI systems.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper establishes a fundamental impossibility theorem: no LLM capable performing non-trivial knowledge aggregation can simultaneously achieve truthful (internally consistent) knowledge representation, semantic information conservation, complete revelation of relevant knowledge, and knowledge-constrained optimality. This 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 in the strictly concave settings the score of an aggregate of diverse beliefs strictly exceeds the sum of individual scores. That gap may quantify the creation of unattributable certainty or overconfidence -- the mathematical origin 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, idealized reasoning strictly preserves semantic content. By demonstrating that hallucination and imagination are mathematically identical phenomena-grounded in the necessary violation of information conservation-this paper offers 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.

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