We propose a self-organizing memory architecture (UMA) for perceptual experience provably capable of supporting autonomous learning and goal-directed problem solving in the absence of any prior information about the agent’s environment. The architecture is simple enough to ensure (1) a quadratic bound (in the number of available sensors) on space requirements, and (2) a quadratic bound on the time-complexity of the update-execute cycle. At the same time, it is sufficiently complex to provide the agent with an internal representation which is (3) minimal among all representations which account for every sensory equivalence class consistent with the agent’s belief state; (4) capable, in principle, of recovering a topological model of the problem space; and (5) learnable with arbitrary precision through a random application of the available actions. These provable properties — both the trainability and the operational efficacy of an effectively trained memory structure — exploit a duality between weak poc sets — a symbolic (discrete) representation of subset nesting relations — and non-positively curved cubical complexes, whose rich convexity theory underlies the planning cycle of the proposed architecture.