Object-centric world models forecast future videos by evolving a set of entity slots, but the variables receiving dynamics supervision are often unconstrained visual features. We introduce \method{}, a mask-grounded soft-Hamiltonian world model that makes its position-like state explicitly depend on slot-owned image support.

A frozen video-slot encoder produces slots and masks; spatial moments of mask-owned support form a canonical state $Q$, temporal differences form $P$, and a learned energy supplies a soft directional bias to a bounded learned increment. Decoder-relevant appearance and identity are stored separately in a causal visual context.

A gated composer and bounded residual then combine this context with the propagated phase state to reconstruct decoder-compatible slots. On OBJ3D, given six observed frames and evaluated over the following 30 frames, \method{} reduces LPIPS by 25.0% and spatial MSE by 33.7% relative to the strongest object-centric baseline.

On CLEVRER, given six observed frames and evaluated over the following ten frames, the corresponding reductions are 14.5% and 18.7%. Horizon-resolved visual and object-state measurements show that the complete model accumulates error more slowly throughout the 30-frame closed-loop rollout.

Project page:https://github.com/moshwm-anon/-moshwm-anon.github.io.