High-Order Barrier Recurrent Cell

1. Exact integration point: use a second-order recurrent or state-space cell with state $x=(q,v)$ and nominal learned acceleration $u_{\mathrm{nom}}=\pi_\theta(x,z)$. Choose $h(x)=1-\|x\|^2/R^2$ to bound hidden activations, or $h(x)=r^2-\|q\|^2$ for a position-like latent state. 2. Pseudocode: compute $\psi_0=h$; for $i=1$ through $p-1$, compute $\psi_i=J_{\psi_{i-1}}f+\alpha_i(\psi_{i-1})$ using automatic differentiation; compute $a=J_{\psi_{p-1}}f$ and $b=J_{\psi_{p-1}}g$; set $c=a+b u_{\mathrm{nom}}+\alpha_p(\psi_{p-1})$; apply $u=u_{\mathrm{nom}}+\max(0,-c/(\|b\|^2+\varepsilon))b^\top$. Advance the cell with Euler or RK2 and backpropagate through the nominal network. 3. Computed from the paper: the recursive barrier variables and terminal inequality. Estimated empirically: Lie derivatives, relative degree, and regions where $\|b\|$ is small. 4. First experiment: compare a GRU, spectral-normalized GRU, and HOCBF-GRU on sequential MNIST with equal parameter counts. Use $p=2$, $h=1-\|x\|^2/R^2$, and perturb inputs to induce hidden-state growth. The quantitative prediction is forward invariance: trajectories initialized with $\psi_0,\psi_1\ge0$ should maintain $h\ge0$ up to numerical error, while the unconstrained GRU should cross $h=0$. The transition should occur when the measured terminal margin $a+b u+\alpha_2(\psi_1)$ changes from negative to nonnegative; after that point, at least 95 percent of violations should disappear.