Residual-screened Koopman latent bottleneck

Integrate this into a recurrent world model or neural state-space model. Use an encoder E_theta(x_t)=z_t in R^m, a linear latent transition z_{t+1}=K_theta z_t, and a decoder D_theta(z_t) for reconstruction. During each training epoch, collect a minibatch of contiguous sequences, form Z_- and Z_+, and compute K_theta=Z_+Z_-^dagger using a differentiable least-squares solve or a periodically refreshed detached solve. Eigendecompose K_theta to obtain eigenvalues lambda_i and right eigenvectors u_i. For numerical stability, normalize each mode denominator with epsilon and combine conjugate-paired modes into real-valued channels. Compute rho_i using the displayed empirical formula over a held-out temporal minibatch rather than only the training batch. Define s_i=1[rho_i<=tau] for hard pruning, or s_i=exp(-rho_i^2/tau^2) for soft weighting, where tau is the 75th percentile of validation residuals during warmup. Forecast by transforming z into the eigenbasis, multiplying retained mode amplitudes by lambda_i at each step, then decoding. The paper supplies the normalized residual criterion; K estimation, eigendecomposition, and tau selection are empirical choices. Start with a 2-layer MLP encoder/decoder and m=32 on Lorenz-63, pendulum, or a small video-prediction dataset. Compare a vanilla latent linear model and an unconstrained recurrent baseline at equal parameter count. Measure one-step MSE, 50-step rollout error, spectral radius, and retained-mode count. Success is lower long-horizon error at equal one-step loss, fewer active modes, and reduced sensitivity to observation noise.