World Models with Predictable Long-Horizon Marginals

Accurate one-step predictions do not ensure that a world model's rollouts retain the data distribution. We make the model's decoded stationary law explicit by learning a decoder of a fixed Gaussian reference and constraining the behaviour-averaged transition to preserve that reference. For controlled systems, a joint transition uses a conditional action chart to preserve behaviour occupancy without requiring invariance at each fixed action. Joint state--action rotations and parallel Gaussian noise give an exactly preserving transition with a tractable conditional density. We derive an absolute convergence bound from finite initialization banks and control departure from the reference through conditional action-space divergence. Across 216 fitted pixel checkpoints on twelve control tasks, the occupancy model with a reference mixture retains every evaluated chain at 105 steps in all 36 task--seed cells, with a rollout-minus-reference energy-statistic difference of -0.0002±0.0003 (training-seed standard error). Each of the four nonpreserving comparison arms loses chains, although the Gaussian arm is more accurate at ten steps. An offline DreamerV3 reference also achieves better short-horizon accuracy. These results distinguish three properties of a world model: the distribution it approaches, the rate of approach, and the conditional dynamics it learns.