{
 "artifacts": null,
 "category": "architecture",
 "description": "Construct a continuous-depth or recurrent residual block with a prescribed polynomial Lyapunov decay near its equilibrium. The architecture combines a fixed radial stabilizer with a learned component that is constrained to have zero radial projection, allowing slow algebraic transients and long memory while preventing asymptotic hidden-state growth.",
 "formulas_latex": [
  "$$dz/dt = -a r^m (z-z_star) + P_perp(z-z_star) h_phi(z,t),   r = ||z-z_star||^2/2$$",
  "$$P_perp(u) = I - uu^T/(||u||^2 + epsilon)$$",
  "$$dr/dt = (z-z_star)^T dz/dt \u003c= -a r^(m+1)$$",
  "$$L_rad = mean(max(0, (z-z_star)^T f_phi(z) + a r^(m+1))^2)$$"
 ],
 "id": 3127,
 "implementation": "Use this in a neural ODE, continuous-depth ResNet, or recurrent state-space block. Set z_star to zero initially and parameterize the hidden dynamics as f_phi(z,t,x) = -a r^m z + P_perp(z) h_phi(z,t,x). For an Euler block, update z_next = z + delta_t f_phi(z,t,x); for longer trajectories use an adaptive ODE solver. At sampled hidden states, compute r and the radial derivative z transpose f_phi(z), then add L_rad to the task loss. A hard alternative is to rescale any proposed update whose empirical radial derivative exceeds the allowed value. Compute the radial quantities exactly on minibatch states; estimate global validity by sampling states, inputs, and small perturbations because the penalty is not a global certificate. The paper supplies the degree-to-decay comparison; the tangent learned field and sampled penalty are the proposed neural adaptation. First test m equal to 0, 1, and 2 against an unconstrained residual MLP and a spectrally normalized residual block on a synthetic long-horizon sequence task and sequential MNIST. After the input is removed, measure hidden-state norms for 100 integration steps and plot r on log-log axes. The prediction is exponential decay for m equal to zero and a log-log slope near -1/m for r when m is positive, with no asymptotic growth provided the radial inequality holds. Increase delta_t to identify the numerical instability boundary and report whether the observed decay law persists below that boundary.",
 "math_summary": "Let z(t) be a hidden state, z_star its equilibrium, r = ||z - z_star||^2 divided by 2, a greater than zero a stabilizing gain, and m greater than or equal to zero a chosen degree. Use the vector field dz/dt = -a r^m (z-z_star) + P_perp(z-z_star) h_phi(z,t), where P_perp(u) = I - uu^T/(||u||^2 + epsilon) approximately removes the radial component of the learned field. For the idealized field without projection error, dr/dt = -a r^(m+1). Thus r decays exponentially when m equals zero and as t^(-1/m) when m is positive; the state norm decays as t^(-1/(2m)). A sampled radial penalty can enforce the same inequality for the learned discretized vector field.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "mlp",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.03120",
  "arxiv_url": "https://arxiv.org/abs/2609.03120",
  "summary_what_math_gives_to_ml": "The paper introduces a quantitative stability degree that distinguishes exponential convergence from algebraic convergence, with degree m greater than zero corresponding to decay proportional to t^(-1/m). Its transferable mechanism is a Lyapunov dissipation inequality whose exponent determines the long-time convergence law, together with direct and converse tests for certifying that exponent. In neural-network training, this can become an adaptive learning-rate controller and an online stability monitor that targets a prescribed decay exponent rather than merely checking whether the loss decreases.",
  "title": "On a Gradation for Asymptotic Stability",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Degree-Calibrated Stable Residual Flow",
 "url": "https://synthcore.org/idea/3127/degree-calibrated-stable-residual-flow",
 "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
   }
  }
 }
}