Latent planners can pick the wrong way: at an acute bend, terminal distance ranks the backward candidate first
Synopsis
When a latent world model ranks candidate actions by terminal Euclidean distance to a goal latent, an acute bend at a mandatory junction can make it prefer a candidate that moves backward over one that makes real progress, even with exact rollouts.
Interpretation
Latent planners score candidate action sequences by the Euclidean distance from their terminal latent to a goal latent, and that score can order candidates incorrectly when the rollout bends sharply. Predictive accuracy was treated as sufficient for reliable terminal-distance guidance; exact agreement with transition distance is stronger than planning needs, since it forces every shortest latent rollout to be a straight segment. A six-branch star study compares a 2D regular hexagon, the same hexagon zero-padded into 3D, and a 3D cross-polytope under one protocol; the first two share identical pairwise angles and six acute pairs, while the cross-polytope has none.
Planning only needs an ordering property: moving forward along a shortest rollout must not make the same future goal appear farther away. This pathwise transition-order consistency is shown to be exactly the classical self-approaching condition, whose differential form keeps future points inside the current forward tangent half-space, so nonzero curvature alone is not a failure. After polygonal interpolation of the discrete latent rollout, a two-segment shortest rollout preserves pathwise order for every source and goal radius if and only if the outward angle is non-acute; an acute outward angle produces a strict reversal.
Latent dimension then reads as angular capacity: if locally distinct transition branches must be pairwise non-acute, their number is bounded by a spherical-code limit. Dimension is no longer just a coordinate count but the number of geometrically separated transition branches that can coexist without acute configurations. The 3D zero-padded control matches 2D at every horizon, while the 3D cross-polytope succeeds, separating available angular arrangement from ambient dimensionality.
Perspective
This report is for researchers and engineers working on latent planning and representation learning, and it concerns the ordering guarantee of terminal Euclidean distance in graph-like point bottlenecks with locally reversible branches. The guarantee is pathwise: it constrains order along the shortest rollout prefixes the planner actually acts on, not global ranking across unrelated paths. The source states no next experiment or deployment recommendation.
The guarantee covers pathwise ordering only, not global ranking across unrelated paths, and it does not imply impossibility for continuous-action tasks. The acute-bend error vanishes once the horizon is long enough to see beyond the local trap. The experiment is a controlled six-branch star, and locally reversible branches are a premise.