Subspace-Restarted State-Space Dynamics

1. Integration point: implement this in a GRU, vanilla RNN, or diagonal state-space model for sequence forecasting or world-model rollouts. Partition the recurrent vector into h in R^d_h, which is always propagated, and c in R^d_c, which carries short-lived contextual information. Use a learned vector c_ref, or zero initially. 2. Pseudocode: compute (h,c_pre) = cell(h,c,u_t); set c = c_pre; if (t+1) modulo r equals zero, set c = (1-rho)c_ref + rho*c_pre; decode from concatenate(h,c). Sweep r in {1,2,4,8,16,32} and rho in {0.25,0.5,0.75,1}. 3. Computed versus estimated: the restart map is exact; estimate drift v and diffusion D from rollout means and variances, and estimate fast-state autocorrelation from hidden trajectories. 4. First experiment: train matched GRUs on a synthetic constant-velocity-plus-noise forecasting task and Sequential MNIST, comparing no restart, full-state reset, and subspace restart. The prediction is a crossover near n_star approximately r: for n below r, variance can be correlated or ballistic, while for n much larger than r, log variance versus log horizon approaches slope 1. Increasing r should move n_star proportionally, with measured n_star divided by r remaining within 20 percent.