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On the Fundamental Impossibility of Hallucination Control in Large Language Models

Created by
  • Haebom

Author

Micha{\l} P. Karpowicz

Outline

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.

Takeaways, Limitations

Takeaways:
LLM provides a fundamental understanding of the problem of hallucinations by revealing that hallucinations are a mathematically inevitable phenomenon, not an engineering problem.
By clarifying the trade-offs between truthfulness, knowledge utilization, and response completeness, we provide a principled foundation for developing hallucination management strategies.
It suggests a new direction for AI research by connecting various fields such as neural network inference, philosophy, game theory, and information theory.
Limitations:
This paper proves the mathematical impossibility of completely eliminating hallucinations, but does not present a specific methodology for effectively managing them.
The mathematical framework used in the proof is complex and may be difficult for general AI researchers to understand.
It may not be possible to present a general mathematical model that fully encompasses the hallucinatory phenomena of real LLM.
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