Reducible cube dynamics on regular contractible sets in four dimensions

We consider the topology-preserving cube dynamics in the higher-dimensional extension of the Ising-disk model. The states are all finite regular contractible unions of lattice cubes containing a fixed cube. A removal must admit a strong deformation retraction that keeps the removed cube inside itself, and additions are the inverse moves. In four dimensions, every state is a piecewise-linear manifold with boundary, and every permitted move preserves the homeomorphism type of that boundary. A contractible four-manifold with nontrivial boundary fundamental group, embedded by Lickorish in the four-sphere, can be realized exactly as such a lattice-cube union by a hypersurface cubulation theorem. It therefore cannot communicate with the state consisting of the fixed cube alone. This gives a negative answer to the proposed universal higher-dimensional extension. The obstruction holds for every choice of positive transition rates; the separate three-dimensional question is not settled here.