We tackle the challenge of building real-time interactive video worlds with user-defined behaviors. Such worlds must track object states, apply interaction rules, and render the resulting consequences as users act. Yet existing end-to-end video world models typically learn interaction dynamics within the generator, making it difficult to specify object-level rules and reason about their consequences. In this work, we introduce BoxWorld, which separates interaction reasoning from generative rendering. Given user-specified rules, a coding agent creates a Box World Engine that determines interaction outcomes and exposes the resulting object states through a lightweight interface of layout boxes, persistent object attributes, and dynamic status variables, leaving appearance and physical effects to generative rendering. A Box-State Generative Renderer then renders this evolving state stream into video autoregressively, enabling online control over object motion, state switches, and conditional events. Rendering this state stream is non-trivial: the renderer must respond to object-level state updates in real time while remaining coherent with the generated history. To this end, we introduce a three-stage training strategy that learns object grounding from open-domain videos and state transitions from gameplay videos, both annotated automatically, and finally performs rollout distillation on self-generated histories. We further propose PairedStateBench, a paired benchmark for fine-grained interaction control. BoxWorld substantially outperforms end-to-end world models and remains competitive across multiple public benchmarks. Project page: https://boxworld-web.github.io.

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