{
 "artifacts": null,
 "category": "dynamics",
 "description": "Train a latent state-space neural network so that its effective pole geometry remains consistent when identified by low-frequency moments and finite-window trajectories. Penalize disagreement between the two reductions, and penalize proximity to the oscillatory/non-oscillatory boundary, to reduce spurious ringing after distillation or context truncation.",
 "download_zip": "https://synthcore.org/download/1153",
 "formulas_latex": [
  "$$p(s)=s^2+a_1s+a_0,\\qquad \\Delta=a_1^2-4a_0$$",
  "$$\\rho_{\\mathrm{moment}}=4.50,\\qquad \\rho_{\\mathrm{window}}=3.31,\\qquad \\rho_{\\mathrm{boundary}}=4$$",
  "$$\\mathcal{L}_{\\mathrm{pole}}=\\left|\\rho_{\\mathrm{moment}}-\\rho_{\\mathrm{window}}\\right|+\\lambda_m[ m-|\\rho_{\\mathrm{moment}}-4| ]_+ +\\lambda_m[ m-|\\rho_{\\mathrm{window}}-4| ]_+$$",
  "$$z_{t+1}=F_\\theta(z_t,u_t),\\qquad J_t=\\frac{\\partial F_\\theta}{\\partial z_t},\\qquad s_i=\\frac{1}{\\Delta t}\\log\\lambda_i(J_t)$$"
 ],
 "id": 3003,
 "implementation": "Integrate this into a small linear or locally linear state-space network, gated RNN, or latent world model. After each validation rollout, collect the latent trajectory and fit the same two-state reduced model in two ways. First, match low-frequency response moments, such as DC gain and the first derivative of the transfer function at zero frequency. Second, fit the reduced model by least squares over a finite trajectory window of length W. Extract both pole coordinates and add their absolute disagreement to the task loss. Pseudocode is: `traj = rollout(model, inputs); model_m = fit_moment_reduction(traj); model_w = fit_window_reduction(traj,W); rho_m = pole_coordinate(model_m); rho_w = pole_coordinate(model_w); loss = task_loss + lam*abs(rho_m-rho_w)`. The reduction procedures and boundary rho = 4 come from the paper; trajectories, Jacobians, fit uncertainty, and local poles are estimated empirically. The first cheap experiment should use a diagonal-plus-low-rank SSM on Copy-Memory and damped-sine sequence prediction, comparing ordinary training with this penalty. The falsifiable prediction is a sharp increase in ringing and long-horizon error when either fitted rho crosses 4. The regularizer should reduce absolute protocol disagreement by at least 50 percent, while the measured oscillation transition should occur within 20 percent of rho = 4.",
 "math_summary": "For a two-state reduced drift with characteristic polynomial p(s) = s^2 + a1 s + a0, the pole discriminant is Delta = a1^2 - 4 a0. Delta less than zero gives a complex-conjugate pair, while Delta greater than or equal to zero gives real poles. The paper reports rho_moment = 4.50 and rho_window = 3.31 for the same higher-order real-relaxation system, with a pole-class boundary at rho = 4. For a neural transition z_(t+1) = F_theta(z_t,u_t), estimate a local Jacobian J_t = partial F_theta / partial z_t, obtain local poles from s_i = log(lambda_i(J_t))/dt, and fit the reduced pole coordinate rho using both reduction protocols.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "spectral-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "world-model",
  "distillation"
 ],
 "paper": {
  "arxiv_id": "2609.01329",
  "arxiv_url": "https://arxiv.org/abs/2609.01329",
  "summary_what_math_gives_to_ml": "The paper shows that pole geometry and latent-state properties can change under model reduction: the same higher-order system of real relaxation modes produces rho_moment = 4.50 under low-frequency moment matching and rho_window = 3.31 under finite-window fitting, crossing the reduced second-order boundary at rho = 4. This mechanism transfers to recurrent and state-space neural networks, where compression or finite-context fitting can create spurious oscillatory modes and degrade long-horizon prediction. The paper also shows that an observed scalar spectrum can be insensitive to hidden cross diffusion when the hidden state has no self-relaxation, even though path-space irreversibility changes. These results motivate protocol-robust pole regularization and latent irreversibility diagnostics.",
  "title": "Pole-Zero Geometry, Model Reduction, and Identifiability in Sensory Adaptation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Reduction-Robust Pole Regularization",
 "url": "https://synthcore.org/idea/3003/reduction-robust-pole-regularization",
 "verification": {
  "peer_reviewed": false,
  "status": "queued_mechanism",
  "status_label": "Queued — mechanism check",
  "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
   }
  }
 }
}