Normally Hyperbolic Latent Attractor

1. Integration point: use this as the transition module of a GRU-free recurrent model, latent ODE discretization, or world model. Split the hidden state as $z=(u,v)$ with $u\in\mathbb{R}^k$ and $v\in\mathbb{R}^{d-k}$. Implement $u_{t+1}=f_\theta(u_t,v_t)$ and $v_{t+1}=\alpha h_\theta(u_t,v_t)$, where $h_\theta$ is spectrally normalized in its $v$ input and $\alpha<1$ is fixed or learned through $\alpha=\sigma(a)$. 2. Pseudocode: initialize $u,v$; compute $(u',v')=F_\theta(u,v)$; add the rollout loss; on randomly sampled manifold states $(u,0)$ compute $r_{inv}=\|g_\theta(u,0)\|^2$; estimate the transverse Jacobian $A(u)$ with Jacobian-vector products and power iteration; add $r_{NH}=\max(0,\widehat{\sigma}_{max}(A(u))-q)^2$; update $\theta$ using $L=L_{task}+\lambda_{inv}r_{inv}+\lambda_{NH}r_{NH}$. 3. The invariant-manifold condition and contraction inequality are taken from the NHIM mechanism. $A(u)$, its largest singular value, and the decay rate are estimated by automatic differentiation; a full Jacobian is unnecessary. Monitor the tangent Jacobian $B(u)$ but do not force it below one, allowing oscillatory or slowly varying modes. 4. First cheap experiment: train a 2-layer latent RNN on noisy Lorenz-63 one-step and 50-step prediction, comparing a standard RNN with the split model at equal hidden dimension. Use $k=2$, $d=8$, and $q=0.8$. Perturb only $v_0$ while holding $u_0$ fixed. The quantitative prediction is $\|v_t\|/\|v_0\|\approx q^t$ until nonlinear saturation; the fitted log-norm slope should be near $\log(0.8)=-0.223$, with held-out transverse singular values below $0.8$. Setting the target above the contraction boundary, $q>1$, should yield growing off-manifold errors and worse long-horizon stability, while tangent prediction quality should change much less.