This paper presents a fundamental impossibility theorem, stating that a large-scale language model (LLM) capable of processing non-obvious knowledge sets cannot simultaneously achieve truthful knowledge representation, semantic information preservation, complete disclosure of relevant knowledge, and knowledge-constrained optimization. This impossibility stems not from an engineering limitation, but from the mathematical structure of the information set itself. The paper demonstrates this by describing the inference process as an idea auction among distributed components competing to form responses using partial knowledge. The proof spans three independent mathematical areas: mechanism design theory (Green-Laffont), appropriate scoring rule theory (Savage), and direct structural analysis of transformers (Log-Sum-Exp convexity). Specifically, we demonstrate how to quantify the generation of overconfident or intuitive responses (characteristics of hallucinations, creativity, or imagination). To support this analysis, we introduce complementary concepts of semantic information measures and emergence operators to model constrained inference in general settings. We demonstrate that while constrained inference generates accessible information that provides valuable insights and inspiration, ideally unconstrained inference strictly preserves semantic content. By demonstrating that hallucinations and imagination are mathematically equivalent phenomena based on their deviations from truthfulness, semantic information preservation, relevant knowledge disclosure, and knowledge-constrained optimization, we provide a principled basis for managing these behaviors in advanced AI systems. Finally, we offer some conjectural ideas for evaluating and improving the proposed theory.
