Topological Necessities: Mechanism-Invariant Strategic Subgoals for Cross-Embodiment Goal-Conditioned Control

Abstract:Long-horizon goal-conditioned reinforcement learning delegates control to a high-level module that proposes subgoals, but existing subgoals are implicit byproducts of value functions or latent actions, tied to the executor that produced them. We study a different object: a route-conditioned order of unavoidable stages that every successful executor must traverse, recoverable from offline trajectories and belonging to none of them. Its defining properties are topological: an unskippable stage is a separating set that every admissible path must cross, and a loop in free space forces a route choice. We read the two by homology in dimensions 0 and 1 over a transport-weighted carrier built from successful trajectories, yielding an enumerable gate set with shell-level certificates; the certified gates are what we call topological necessities. Certified gates enter the decision loop as a recursive topological gate hierarchy. Under a fixed, isomorphic free space, the object survives executor replacement: gates frozen on PointMaze data transfer without retraining to Ant and Humanoid, attaining the highest Humanoid aggregate under a unified interface (96.1), with +36.0 over a map-privileged reference on the multi-route task (p=1.4e-5); the planner saturates PointMaze (100+/-0) and matches or exceeds the strongest baselines on AntMaze (giant +22.9) and Kitchen (+15.8/+12.6).

60 pages total (9-page main text + appendices), 16 figures. Code and data: this https URL

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Robotics (cs.RO)

arXiv:2609.11014 [cs.LG]

(or arXiv:2609.11014v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.11014 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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