Sobolev-Certified Conditional Operator

1. Exact integration point: apply this to a neural stochastic state-space model, world model, or learned Koopman predictor. Use a conditional density head $q_\theta(y\mid x)$, preferably a Gaussian mixture or discretized likelihood so derivatives with respect to $x$ are available by automatic differentiation. For high-dimensional $y$, use a random-feature approximation to the conditional mean embedding rather than evaluating the full density. 2. Pseudocode: sample minibatch $(x_i,y_i)$; compute negative log likelihood; draw $M$ reference outcomes or quadrature points $\tilde y_m$; use autodiff to compute $\partial_x^\alpha q_\theta(\tilde y_m\mid x_i)$ for all multi-indices $|\alpha|\le s$; form the Monte Carlo estimate of $E_s$; construct an empirical matrix $A_{ab}=\langle\psi_a,K_\theta\phi_b\rangle$ using random Fourier or fixed Sobolev basis functions; penalize squared singular values above budget $B^2$; backpropagate the total loss. 3. Computed from the paper: the CEO definition, Sobolev regularity criterion, and Hilbert–Schmidt interpretation. Estimated empirically: density derivatives, integrals over $y$, operator singular values, and budget $B$, selected from validation data or an unregularized baseline. 4. First cheap experiment: train a 2D stochastic nonlinear dynamical-system model on trajectories from $x_{t+1}=0.8x_t+0.1\tanh(x_t)+\epsilon_t$, with Gaussian noise, comparing an MLP conditional-Gaussian baseline against the same model with $s=1$ regularization and a singular-value penalty. Measure one-step likelihood, 50-step rollout error, derivative energy, and the empirical Koopman singular-value spectrum. The quantitative prediction is that increasing $\lambda_s$ monotonically reduces estimated Sobolev energy and yields a faster-decaying singular-value tail; rank-$r$ truncation error should match the measured tail $\sum_{j>r}\sigma_j^2$. Falsify the idea if energy does not decrease or if truncation and rollout errors do not improve at matched likelihood.