Learning Rules, Memorizing Facts: An Information-Theoretic View of Hallucination

Understanding the relationship between generalization and hallucination requires clarifying the role of memorization. We study this relationship from an information-theoretic perspective through a Selector-Density Conditional Mutual Information (CMI) framework. In the sparse random-fact limit, a uniformly bounded information budget per fact forces a nonvanishing false-positive rate at recall bounded away from zero, even when all facts are observed during training. In a rules-and-facts setting, we decompose model-data mutual information into rule learning, atomic-fact memory, and excess memorization, revealing competition between rule learning and factual memory even without excess memorization. A joint budget bound quantifies the resulting constraint on atomic hallucination under perfect atomic recall. Moreover, an additive Gaussian model shows that maintaining high recall for atomic and composite facts incurs false positives even under optimal data mixing. Finally, controlled relational experiments hint how training exposure, rule complexity, and model capacity shape these trade-offs.