Lyapunov-sign-preserving neural time stepping

Use this as the solver and safety controller for a stochastic neural ODE, latent world model, or continuous-time RNN. The network supplies $f_\theta$ and $G_\theta$; initially leave the training objective unchanged and expose the integration step $h$ as a runtime control variable. Implement Euler-Maruyama and a second, more stable update such as a semi-implicit drift step. Initialize a normalized random probe $q$. For each transition compute $z_{k+1}=\Phi_h(\omega_k,z_k)$, obtain the Jacobian-vector product $v=J_kq$ using automatic differentiation, accumulate $s\leftarrow s+\log(\|v\|+10^{-12})$, and set $q\leftarrow v/(\|v\|+10^{-12})$. Every $M$ steps compute $\widehat\lambda_h=s/(Mh)$. If the estimate is close to zero, halve $h$; if a supposedly stable mode has $\widehat\lambda_h>0$, halve $h$ or switch to the stable solver. Jacobian-vector products are computed exactly; the exponent, convergence order $p$, and constant $C$ are estimated empirically using runs at $h$, $h/2$, and $h/4$. First experiment: train a two-dimensional stochastic neural ODE on noisy damped-oscillator rollouts, compare Euler-Maruyama with the controlled solver, and sweep $h$. The prediction is that the exponent sign becomes invariant under refinement and that $|\widehat\lambda_h-\widehat\lambda_{h/2}|$ scales as $h^p$; any coarse-step sign flip should disappear after halving $h.