Freezing the Geometry, Adapting the Dynamics: Minimum Action Distance in Non-Stationary Environments

The Minimum Action Distance (MAD) measures the minimum number of decision steps between two states and, by definition, depends on the transition kernel *only through its support*. It should therefore survive arbitrary changes to transition probabilities that leave reachability intact. We construct a two-path corridor in which a region of reduced action-execution probability moves between the two branches while the transition support stays identical, freeze a MAD encoder trained once, and learn only a cheap latent state-conditioned action model $f_\psi(z,a)\approx\mathbb{E}[z'-z \mid z,a]$ on top. A latent model-predictive planner with a horizon 3.7 times shorter than the distance to the goal re-routes after every one of 21 relocations, while the same planner with a frozen action model never re-routes in 12 of them.