{
 "artifacts": null,
 "category": "theory",
 "description": "Train a neural state-space model whose one-step dynamics are linear in a fixed analytic feature vector, and use the empirical feature Gram matrix to detect whether passive trajectories identify the dynamics. Add data collection or replay only when the Gram matrix is poorly conditioned; the analytic-feature assumption predicts that persistent excitation should emerge without deliberately visiting every operating mode.",
 "formulas_latex": [
  "$$x_{t+1}=\\Theta_*\\phi(x_t,u_t)+w_t$$",
  "$$G_N=\\frac{1}{N}\\sum_{t=0}^{N-1}\\phi(x_t,u_t)\\phi(x_t,u_t)^\\top,\\qquad \\widehat{\\Theta}_N=\\left(\\sum_{t=0}^{N-1}x_{t+1}\\phi_t^\\top\\right)\\left(\\sum_{t=0}^{N-1}\\phi_t\\phi_t^\\top+\\lambda I\\right)^{-1}$$",
  "$$\\|\\widehat{\\Theta}_N-\\Theta_*\\|_F\\ \\lesssim\\ \\sigma\\sqrt{\\frac{n_xp\\log(1/\\delta)}{N\\lambda_{\\min}(G_N)}}$$",
  "$$a^\\top\\phi(z)=0\\ \\text{on an open set}\\ \\Longrightarrow\\ a^\\top\\phi(z)=0\\ \\text{on the connected analytic domain}$$"
 ],
 "id": 2856,
 "implementation": "1. Exact integration point: use a neural state-space or world-model predictor with $\\hat x_{t+1}=\\Theta\\phi_\\psi(x_t,u_t)$, where $\\phi_\\psi$ is either a fixed analytic basis such as polynomial, Gaussian RBF, or sine/cosine features, or a small analytic MLP using smooth activations such as tanh. Keep the final dynamics layer linear in the features. 2. Pseudocode: collect passive rollouts; compute $\\phi_t=\\phi_\\psi(x_t,u_t)$; update $G\\leftarrow(1-\\alpha)G+\\alpha\\phi_t\\phi_t^\\top$; solve the ridge least-squares update for $\\Theta$; backpropagate prediction loss into $\\psi$ only if desired; if $\\lambda_{\\min}(G)\u003c\\gamma\\operatorname{tr}(G)/p$, add a small bounded perturbation to actions or prioritize replay samples from underrepresented feature directions. 3. Computed quantities are $G$, its eigenvalues, the least-squares parameters, and prediction residuals. The paper supplies the analytic identifiability mechanism and finite-sample convergence principle; constants, noise assumptions, and the threshold $\\gamma$ must be estimated empirically. 4. First cheap experiment: identify a two-dimensional nonlinear pendulum or Duffing simulator from passive Gaussian action noise using polynomial-feature, RBF-feature, and ReLU-feature models. Compare parameter error and long-horizon rollout error versus sample count while plotting $\\lambda_{\\min}(G_N)$. The prediction is that analytic polynomial and RBF models show eventual rank growth and error decay approximately proportional to $[N\\lambda_{\\min}(G_N)]^{-1/2}$, while a smooth bump feature that is exactly zero in the visited region exhibits persistent Gram-rank failure despite dense passive data. A practical acceptance criterion is a strong correlation between decreasing parameter error and increasing normalized minimum eigenvalue, with failure whenever $\\lambda_{\\min}(G_N)/\\operatorname{tr}(G_N)\u003c10^{-3}$ for many consecutive windows.",
 "math_summary": "The modeled dynamics are $x_{t+1}=\\Theta_*\\phi(x_t,u_t)+w_t$, where $x_t$ is the state, $u_t$ is the input, $w_t$ is noise, $\\phi:\\mathbb{R}^{n_x+n_u}\\to\\mathbb{R}^p$ is a known real-analytic feature map, and $\\Theta_*\\in\\mathbb{R}^{n_x\\times p}$ is the unknown parameter matrix. Given samples $z_t=(x_t,u_t)$, define the empirical Gram matrix $G_N=\\frac{1}{N}\\sum_{t=0}^{N-1}\\phi(z_t)\\phi(z_t)^\\top$. Least squares uses $\\widehat{\\Theta}_N=(\\sum_t x_{t+1}\\phi(z_t)^\\top)(\\sum_t\\phi(z_t)\\phi(z_t)^\\top+\\lambda I)^{-1}$, with ridge parameter $\\lambda\\ge 0$. The identification-relevant certificate is $\\lambda_{\\min}(G_N)\u003e0$; under sub-Gaussian noise of scale $\\sigma$, parameter error is expected to decrease on the scale $\\sigma\\sqrt{p/(N\\lambda_{\\min}(G_N))}$. The analytic mechanism is that if a linear combination $a^\\top\\phi(z)$ vanishes on an open connected region, analyticity implies it vanishes throughout the connected domain; non-analytic smooth features do not have this property. Thus passive data should show increasing Gram rank, while a persistent near-zero eigenvalue is a falsifiable failure signal.",
 "math_tags": [
  "dynamical-systems",
  "statistics",
  "linear-algebra",
  "approximation-theory",
  "control-theory"
 ],
 "ml_areas": [
  "world-model",
  "optimizer",
  "training-dynamics",
  "rl"
 ],
 "paper": {
  "arxiv_id": "2608.29908",
  "arxiv_url": "https://arxiv.org/abs/2608.29908",
  "summary_what_math_gives_to_ml": "The paper's key mechanism is that real-analytic feature maps make passive, non-active exploration sufficient for finite-sample identification, whereas merely smooth or piecewise-affine features can remain indistinguishable on the visited region. The transferable asset is an identifiability certificate based on the feature Gram matrix: passive trajectories should eventually produce a nonsingular Gram matrix because a nonzero analytic feature combination cannot vanish on an open set without vanishing globally. For neural state-space models and world models, this suggests using analytic feature bases or analytic adapters together with a Gram-conditioning monitor and an exploration/data-collection schedule that stops only after the smallest Gram eigenvalue crosses a target threshold.",
  "title": "Finite Sample Identification of Analytic Nonlinear Systems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "sample-efficiency",
  "accuracy"
 ],
 "title": "Analytic Passive-Identification Monitor",
 "url": "https://synthcore.org/idea/2856/analytic-passive-identification-monitor",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}