Hurwitz Latent Observer

1. Integration point: use this observer as the hidden-state update in a continuous-time neural SSM or neural ODE. At each observation interval, integrate the learned flow and apply either continuous correction $G(y-Cz)$ or the discrete update $z\leftarrow z+\Delta t[f_{\theta}(z)+G(y-Cz)]$. Let $C$ select observed latent coordinates or represent a fixed sensor projection. 2. Pseudocode: initialize $z$; for each minibatch and time step compute $r=y-Cz$, compute $v=f_{\theta}(z)+Gr$, and update $z\leftarrow z+\Delta t v$. Compute Jacobian-vector products for $J_{\theta}(z)$, form $H=U_K^{\mathsf T}(J_{\theta}-GC)U_K$, estimate its largest symmetric eigenvalue, and add the penalty $\beta\max(0,\lambda_{\max}(H^{\mathsf T}P+PH)+2\alpha\lambda_{\min}(P))$. Parameterize $P=LL^{\mathsf T}+\epsilon I$. 3. Exact quantities are the observer equation, $C$, $K$, and the Lyapunov inequality; Jacobians and the compact state-range coverage are estimated empirically. 4. First experiment: train a two-layer neural ODE or GRU on Lorenz-63 trajectories while observing only two coordinates. Compare a standard model, residual nudging without the penalty, and this observer. Sweep gain and integration step. The predicted signature is a sharp boundary: when the maximum projected symmetric eigenvalue becomes positive, reconstruction error stops decaying; when it is negative by margin $\alpha$, the log-error slope should be approximately $-\alpha$, within 20 percent.