Same States, Different Actuators: What Single-Actuator Data Must Cover for Compositional World Models

Compositional world models aim to predict unseen actuator combinations without collecting data for every combination. For torque-driven systems, the joint response at a fixed state is the sum of single-actuator responses up to a residual of 1.7–13.6% of the actuated response, so single-actuator data suffice if each actuator's response is known at the states the combination visits. Reset-only collection observes each actuator only at states reached by that actuator alone from reset, and existing coverage measures cannot flag this: concentrability is unbounded for joint actions under any single-actuator data, and state coverage ignores which actuator is observed at which state, the dataset's actuator–state pairing. A matched-pool intervention changes which actuator is executed at each state of a fixed pool, keeping per-actuator action marginals fixed, and so isolates the pairing. In a linear switched class, we prove that histories engaging only a proper subset of actuators, as reset-only histories do, can leave joint behaviour undetermined even though together they reach every state, and we give a necessary and sufficient condition for component observations to determine it. A collector must therefore let a history engage several actuators; state-carrying collection does so by switching actuators without resetting. Across four MuJoCo environments and two model classes, balanced pairing reduces one-step prediction error on unseen combinations by 8.3–29.2%, and by 8.4–25.4% when the joint prediction is assembled from the model's own single-actuator responses. At the same interaction budget, state-carrying collection reduces one-step error by 6.3–34.6% relative to reset-only collection, and its advantage extends to 120 and 98 of 121 unseen actuator subsets for the two model classes and to ten-step rollouts. The artifacts are available at https://figshare.com/s/2aec1f48b5d5a038aca6.