{
 "artifacts": null,
 "category": "dynamics",
 "description": "Build a recurrent or state-space model with latent coordinates split into tangent variables that carry the modeled dynamics and transverse variables that contract toward a learned invariant manifold. Penalize violations of invariance and enforce a spectral gap between tangent and transverse Jacobian dynamics, preserving useful noncontracting behavior while suppressing off-manifold drift during long rollouts.",
 "formulas_latex": [
  "$$F_\\theta(u,v)=\\begin{bmatrix}f_\\theta(u,v)\\\\g_\\theta(u,v)\\end{bmatrix},\\qquad \\mathcal{M}=\\{(u,v):v=0\\},\\qquad g_\\theta(u,0)=0.$$",
  "$$A_\\theta(u)=\\left.\\frac{\\partial g_\\theta}{\\partial v}\\right|_{v=0},\\qquad \\sup_{u\\in\\mathcal{M}}\\|A_\\theta(u)\\|_2\\le q\u003c1.$$",
  "$$\\|v_t\\|\\le Cq^t\\|v_0\\|\\quad\\text{for sufficiently small }\\|v_0\\|.$$",
  "$$\\dim W^s(\\mathcal{M})=\\dim\\mathcal{M}+\\dim E^s.$$"
 ],
 "id": 3136,
 "implementation": "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 $\\alpha\u003c1$ 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, $q\u003e1$, should yield growing off-manifold errors and worse long-horizon stability, while tangent prediction quality should change much less.",
 "math_summary": "For a discrete latent map $z_{t+1}=F_\\theta(z_t)$, let the intended invariant manifold be $\\mathcal{M}=\\{(u,v):v=0\\}$, where $u\\in\\mathbb{R}^k$ are tangent coordinates and $v\\in\\mathbb{R}^{d-k}$ are transverse coordinates. Invariance requires $F_{\\theta,v}(u,0)=0$. Normal hyperbolicity is approximated by requiring the transverse Jacobian $A(u)=\\partial_vF_{\\theta,v}(u,0)$ to contract uniformly, $\\|A(u)\\|_2\\le q\u003c1$, while tangent dynamics $B(u)=\\partial_uF_{\\theta,u}(u,0)$ may have larger or near-unit singular values. If $v_{t+1}=A(u_t)v_t+O(\\|v_t\\|^2)$, then $\\|v_t\\|\\lesssim q^t\\|v_0\\|$ locally. The stable-manifold dimension mechanism is $\\dim W^s(\\mathcal{M})=\\dim\\mathcal{M}+\\dim E^s$, where $W^s(\\mathcal{M})$ is the set of states asymptotic to the manifold and $E^s$ is the stable transverse bundle.",
 "math_tags": [
  "dynamical-systems",
  "geometry",
  "control-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2609.03257",
  "arxiv_url": "https://arxiv.org/abs/2609.03257",
  "summary_what_math_gives_to_ml": "The paper provides a constructive trapping mechanism based on a normally hyperbolic invariant manifold (NHIM): a lower-dimensional central waist is invariant, tangent dynamics may remain neutral, while transverse dynamics are hyperbolic and contract trajectories toward it. The stable-manifold theorem predicts a family of states that asymptotically approaches the manifold, while recurrence prevents an open trapped set in the full phase space. This transfers naturally to recurrent neural networks and learned world models by separating latent coordinates into persistent tangent variables and contracting transverse variables. The key falsifiable benefit is controlled long-horizon behavior: transverse perturbations should decay geometrically at a rate predicted by the learned Jacobian without forcing all useful tangent dynamics to contract.",
  "title": "On the trapping of ray families by mirrors",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Normally Hyperbolic Latent Attractor",
 "url": "https://synthcore.org/idea/3136/normally-hyperbolic-latent-attractor",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}