Support-Sparse Koopman World Model

Integrate this into a small state-space or world-model network, not into a language model. Use an MLP encoder E_theta mapping state vectors to a latent width d_z such as 32 or 64, an MLP decoder D_phi, and a trainable matrix K. Training batches should contain contiguous windows (x_k,...,x_{k+H}) from trajectories; use H=5 initially and increase to 10 after reconstruction stabilizes. For each window, compute z_k=E_theta(x_k), decode it, compute the one-step latent residual E_theta(x_{k+1})-K z_k, recursively generate K^h z_k, decode every rollout, and add the L1 latent penalty. Compute s(z) with absolute value divided by absolute value plus epsilon, using epsilon=10^{-6}, and add the support-persistence penalty for adjacent states from the same trajectory. Pseudocode: z=E(x[:,0]); rec=mean((D(z)-x[:,0])^2); pred_lat=0; pred_obs=0; sparse=mean(abs(z)); support=0; for h in 1..H: z=K @ z; pred_lat += mean((z-E(x[:,h]))^2); pred_obs += mean((D(z)-x[:,h])^2); support += mean(abs(s(E(x[:,h]))-s(E(x[:,h-1])))); loss=rec+alpha*pred_lat/H+gamma*pred_obs/H+lambda*sparse+eta*support. Select tau on validation data to produce an average active fraction of 10 to 30 percent, then cluster binary masks to test basin separation. The first experiment should use a two-basin nonlinear system such as a double-well oscillator or a mixture of two stable systems, with 1,000 training trajectories and held-out trajectories from each basin. Compare against a dense Koopman autoencoder with identical encoder width, parameter count, optimizer, and rollout loss. Success requires lower 20-step state MSE, less long-rollout divergence, and support-mask classification accuracy above the dense baseline on held-out basin-interior states. Include ablations without support persistence and with only one-step loss; the method is supported only if sparse supports remain stable along trajectories and forecasting improves at equal latent width.