Do World Models Learn Global Understanding?

AI systems often feel frustratingly brittle and fragmented. A large language model (LLM) may correctly explain a concept but fail to apply it, or follow safety instructions in one context but not another. This behavior suggests a general failure to lift local information to a global understanding. To gain fundamental insight, we frame "understanding" as learning constraints and propagating their consequences. We construct learning tasks on monoid worlds, sets of states connected by action transitions (state, action, next state), where observed training transitions and an underlying unseen constraint jointly determine held-out transitions. Measuring generalization tests whether models can learn global constraints from local transitions and propagate their consequences. We consider inverse, commutativity, composition, and periodicity constraints relevant to spatial and semantic structure. Across attention, recurrent, and state-space architectures, next-state training on paths of observed transitions fits the data but fails to propagate non-trivial constraints. Compositional training, which uses identical paths but hides intermediate states from the input, achieves 96% accuracy on inverse, commutativity, and composition constraints across architectures, yields corresponding improvements in geometric generalization of world models trained on embodied environments (+42% on propagating commutative constraints) and relational generalization in Wikidata-finetuned LLMs (+73% on composition propagation). How far do models propagate constraints when inferring an unseen fact may depend on first inferring others? We define proof depth $d$ of a held-out transition, measuring the minimum number of inference rounds to infer the transition, and find that model generalization decreases sharply with increasing proof depth. A compositionally trained transformer on length $T=2$ paths generalizes at 95% for transitions derivable directly from observations ($d=1$) but only at 49% for those requiring an additional inference round ($d=2$). Increasing compositional path length $T$ improves generalization, and $T=4$ compositional models achieve 95% accuracy on $d=2$ transitions. These results provide a formal way to investigate global understanding in language and world models and demonstrate that compositional training promotes information propagation and integration.