The Cost of Compression: A Rate-Distortion Limit on Factual Hallucination

Factual hallucination in closed-book question answering is often treated as a coverage problem: a model fails because the relevant fact is absent from its internal memory. This view misses a second source of error. Even when a fact has been observed, finite memory may force it to be stored only approximately. We study this effect through a simple coverage--compression model of factual recall. We consider an unstructured question-answering task with N possible queries and K possible answers. A learner observes M training facts, compresses them into at most B bits, and answers uniformly drawn test queries without retrieval. For a uniformly random ground-truth mapping, we prove E at least M over Nδ! (B over M) + (1-M over N) (1-1 over K), where δ(r) is the inverse rate-distortion function of a uniform K-ary source under zero-one loss. The two terms separate compression distortion on observed facts from missing coverage on unobserved facts. The bound gives a compact way to reason about selective memory, forced compression, structure, retrieval, abstention, and long-context organization. We study the predicted signatures with theory-implied simulations and controlled fact-injection probes in modern language models that vary fact load and effective trainable memory. The result is not a complete theory of hallucination, but an information-theoretic account of a separable failure mode: lossy recall of observed facts under finite memory.