Order-Consistent Latent Geometry for Reliable Short-Horizon Planning in World Models

Abstract

Latent planners often rank candidate action sequences by terminal Euclidean distance to a goal latent. Exact agreement with transition distance would make this score ideal, but is stronger than planning requires: along a transition-shortest rollout, the essential requirement is that latent distance does not reverse true transition progress. We show that this pathwise no-reversal requirement is precisely the classical self-approaching condition. To the best of our knowledge, we are the first to identify self-approachingness as a planner-facing compatibility condition for terminal Euclidean latent planning and to connect its violation to an exact finite-horizon candidate-order reversal. Because visual world models predict discrete latent sequences, we analyze their polygonal interpolation. We prove that an acute bend creates a nonempty region in which terminal Euclidean distance ranks a transition-worse candidate ahead of one that makes genuine progress, even under exact rollouts. We further interpret latent dimension as angular capacity through a classical spherical-code bound. A controlled six-branch experiment, including a zero-padded three-dimensional baseline, validates the predicted ordering failure and separates angular arrangement from ambient dimensionality.