Minimal Markovization via Stable Quotients in Holonomy-Cover Decision Processes

In partially observable environments, an agent must maintain a recursively updateable statistic of history to restore the Markov property, but the smallest such statistic is often unknown. This paper focuses on holonomy-cover decision processes, a structured POMDP subclass where visible dynamics are Markov and each visible transition applies a fixed permutation to a hidden mode. The authors construct the stable quotient, the coarsest observation-wise abstraction preserving one-step rewards and quotient successors, and prove that the pair of current observation and stable class forms an exact finite Markov state. When correctly initialized, exact class tracking requires exactly the minimal memory symbols—under reachability and pairwise decision separation, no finite-memory controller can use fewer. With resettable diagnostics, nearest-prototype class inference achieves exponentially decaying error, and a calibrate-then-restart reduction transfers finite-MDP guarantees to the recovered state. The proposed Holonomy Memory Reinforcement Learning framework represents memory by the current stable class, updates it through ordered edge transports, identifies local class coordinates when diagnostics are available, and applies a standard finite-MDP RL backbone after synchronization. Experiments show exact compression from raw states to quotient states, achieving perfect paired-order accuracy with three decision-time memory states, matching the quotient oracle and outperforming non-oracle baselines.