Backward-error penalty for learned latent dynamics

Integrate the penalty after the latent transition matrix in a Koopman-inspired state-space model, recurrent predictor, or world model. For each minibatch of sequences, encode z_t=E_theta(x_t), estimate K_theta by ridge regression K_theta=Z_+Z_-^T(Z_-Z_-^T+eta I)^(-1), and compute its s eigenpairs (lambda_i,u_i), selecting modes with largest latent energy or decoder sensitivity. Form each residual r_i=K_theta u_i-lambda_i u_i and stack them into R; stack eigenvectors into Phi. Compute the pseudoinverse with an SVD Phi=U Sigma V^T, replacing singular values below 1e-5 times the largest singular value by zero, then calculate DeltaK=R Phi^dagger. Add beta times the squared spectral norm, approximated by 3-5 power iterations, to the task or reconstruction loss. Detach eigenvectors and K for the first prototype to avoid unstable eigenvalue gradients; refresh the penalty every 10-50 optimizer steps, then test a fully differentiable version. The exact backward-error relation is computed from the paper's residual construction, while ridge strength eta, mode selection, and power-iteration tolerance are empirical. Compare against ordinary one-step latent MSE, spectral-radius clipping alone, and Jacobian regularization on a noisy Lorenz-63 or Kuramoto benchmark. Evaluate validation residual, ||DeltaK||_2, rollout error at 10, 50, and 100 steps, and degradation under injected observation noise. The key falsifiable signal is lower long-horizon error and smaller noise-induced degradation at matched one-step loss, without simply shrinking all eigenvalues toward zero.