Komuro Time-Warp Expansivity Regularizer

1. Integration point: use a latent neural ODE, controlled differential equation, or continuous-time state-space model with latent state z and vector field f_theta. Add the regularizer to reconstruction or prediction loss after producing two rollouts. Sample negative pairs x,y from different trajectories and positive pairs by setting y=phi_s(x) for a small random time shift s. 2. Pseudocode: integrate z_x(t)=ODESolve(f_theta,z_x(0),grid) and z_y(t); represent a monotone warp as alpha(t)=t+sum_k softplus(a_k)B_k(t), normalized so alpha(0)=0; optimize the warp coefficients for a few gradient steps or use soft-DTW; compute a smooth maximum over aligned distances; apply max(0,m-D)^2 only to negative pairs; update theta with L_task+lambda L_exp. 3. Computed quantities are rollouts, warped distances, and the loss. The theorem's true delta is unavailable, so estimate delta_emp from held-out pairs and vary q, epsilon, and warp complexity. 4. First cheap experiment: train a 2D or 4D neural ODE on a synthetic limit-cycle-plus-saddle system, comparing ordinary contrastive separation with this loss. Measure false closeness of negative pairs over long horizons and prediction error under a twofold clock-speed perturbation. The predicted signature is a sharp empirical transition in false-closeness probability near delta_emp(epsilon), while same-orbit pairs remain close after alignment; the benefit should persist under irregular sampling and should not be explained only by increased latent norm.