Asymptotically Commuting Recurrent Blocks

Use a recurrent, state-space, or deep residual model with time-dependent blocks h_t = f_t(h_(t-1), u_t). For a first test, let f_t be a shared MLP with a low-rank time embedding or learned step-size modulation, making the model genuinely non-autonomous. Add lambda_c times L_comm to the task loss and compute it on hidden states from the current minibatch. Sample r from the first half of a sequence and n from the last quarter, with n different from r. Use normalized hidden states or a bounded projection when evaluating the commutator. Pseudocode is: sample hidden states x and indices n,r; compute a = f_n(f_r(x)); compute b = f_r(f_n(x)); set comm = mean squared error between a and b; optimize task_loss + lambda_c times (n/T)^p times comm. During evaluation, run the normal sequence and locally transposed sequences in which selected update pairs are swapped, then measure prediction disagreement. Compute the two compositions exactly; estimate the supremum c_(n,r) with a maximum or 95th percentile over sampled states. Estimate P_epsilon from validation pairs, but enforce a covariance-eigenvalue floor v_min to prevent trivial representation collapse. First experiment: train a GRU or residual SSM on sequential MNIST, permuted sequential MNIST, and a synthetic switching linear system. Compare the baseline, commutator regularization, and random-order augmentation. Sweep lambda_c. The prediction is that the late-time commutator curve decreases with n, approximately as a power law n^(-alpha) or toward a plateau, and that reordered-versus-normal prediction error decreases with the measured commutator. Require at least a twofold reduction in median late-time commutator and a 20 percent reduction in order-perturbation error without reducing hidden-state covariance rank. If P_epsilon rises while covariance collapses, the experiment falsifies the intended mechanism.