Koopman Preview Gate for Adaptive Neural Computation

First, integrate the mechanism after the hidden state of a GRU, state-space model, or causal Transformer. Train an encoder E_phi from hidden state h_t, a d-dimensional linear Koopman predictor K_theta, a cheap base head B_omega, and an expensive refinement head C_psi. Define risky states using the future base-model loss: fit Q and threshold tau so that R(z_t) predicts whether the base model's loss over the next H tokens or timesteps exceeds a selected quantile. Second, use this pseudocode: compute z=E_phi(h); set risk=0; for j in 1,...,H, replace z by K_theta z and update risk=max(risk,z^T Q z); set g=(risk>tau); compute y_base=B_omega(h); if g is true compute y=y_base+C_psi(h), otherwise set y=rho*y_previous+(1-rho)*y_base. During training, replace the hard gate with sigmoid((risk-tau)/T), anneal temperature T, and add the Koopman loss sum_j ||E_phi(h_{t+j})-K_theta^j E_phi(h_t)||^2. Matrix powers, risk, gating, and interpolation are computed online; E_phi, K_theta, Q, tau, H, and rho are learned or calibrated offline. Estimate Q using inverse covariance or parameterize Q=L L^T. Monitor or project K_theta to satisfy rho(K_theta)<1. The first cheap experiment should use a small GRU on Penn Treebank or a nonlinear dynamical-system forecasting benchmark, comparing base-only, always-on refinement, and Koopman-gated refinement. Sweep tau and H. The predicted signature is a positive warning lead time before future-loss spikes, with compute decreasing approximately with gate activation rate. Accept the method if it removes at least 25% of refinement calls while increasing long-horizon error by less than 5%; additionally test whether latent perturbation norms decay approximately as c alpha^j and whether the measured decay agrees with the spectral-radius estimate.