Reduction-Robust Pole Regularization

Integrate this into a small linear or locally linear state-space network, gated RNN, or latent world model. After each validation rollout, collect the latent trajectory and fit the same two-state reduced model in two ways. First, match low-frequency response moments, such as DC gain and the first derivative of the transfer function at zero frequency. Second, fit the reduced model by least squares over a finite trajectory window of length W. Extract both pole coordinates and add their absolute disagreement to the task loss. Pseudocode is: traj = rollout(model, inputs); model_m = fit_moment_reduction(traj); model_w = fit_window_reduction(traj,W); rho_m = pole_coordinate(model_m); rho_w = pole_coordinate(model_w); loss = task_loss + lam*abs(rho_m-rho_w). The reduction procedures and boundary rho = 4 come from the paper; trajectories, Jacobians, fit uncertainty, and local poles are estimated empirically. The first cheap experiment should use a diagonal-plus-low-rank SSM on Copy-Memory and damped-sine sequence prediction, comparing ordinary training with this penalty. The falsifiable prediction is a sharp increase in ringing and long-horizon error when either fitted rho crosses 4. The regularizer should reduce absolute protocol disagreement by at least 50 percent, while the measured oscillation transition should occur within 20 percent of rho = 4.