Non-Commutative State Tracking with Input-Dependent Low-Rank Updates in Mamba-3
State tracking from sequential observations can require both retaining information and updating it by composing observed operations. We extend Mamba-3's diagonal transition with an input-dependent low-rank reflection term to support noncommutative state tracking, in which the order of operations matters. The rank-one update couples state coordinates along an input-dependent direction, enabling non-diagonal state transitions within a single Mamba-3 block. The extension preserves Mamba-3's exponential-trapezoidal discretization, rotary embeddings (RoPE), and readout. For training, we adapt chunkwise computation to parallelize the proposed recurrence within each chunk. Experiments cover group word problems with discrete inputs and a shell game with continuous observations, in which a policy is trained by behavioral cloning. Among the models selected for their strong performance under fixed timing, the proposed model maintains higher tracking success on longer swap sequences in the shell game with continuous observations and timing jitter. These experiments show that the proposed method achieves high accuracy on the evaluated non-commutative tracking tasks, improving on standard Mamba-3. The extension thus offers a Mamba-3-based approach to non-commutative state tracking.