Hamilton-Jacobi-Bellman theory implies that the optimal goal-conditioned action depends on the goal only through the gradient of the goal-reaching distance at the current state, yet standard online GCRL still conditions the actor on the raw goal -- a signal that is geometrically uninformative when the goal is far from the data distribution. We propose Direction-Conditioned Policies (DCP), a fully online method that decomposes goal-reaching into two components sharing one InfoNCE representation
ψ: a subgoal-scoring step that selects a visited state
zt aligned with the final goal
g in
ψg, and a direction-conditioned actor that consumes the unit direction
dt and magnitude
rt from
ψ(st) to
ψ(zt). The two components train jointly, factor cleanly at deployment (subgoal scoring is removed, while direction conditioning remains with
g in place of
zt), and admit independent modification at the same
(dt,rt) interface. We prove three results. First, direction sufficiency under HJB: the optimal action under control-affine dynamics depends on the goal only through the value gradient. Second, a quantitative bound showing that, under mild conditions on the learned representation and assuming the scoring rule returns an on-path
zt, the actor's conditioning input at training and at deployment coincide up to representation error and geodesic slack. Third, a controllable-subspace characterization of when directional conditioning fails. Across nine environments, DCP improves over Contrastive RL on most final metrics, with the largest gains on manipulation and obstacle-interaction tasks; a qualitative analysis of the learned
ψ-distance landscape shows the contrastive representation behaves as an online quasimetric encoding environment topology, and the single failure case (AntSoccer) localizes to a learned-gradient pathology that the theory anticipates.