# Эксперимент: Delay-Margin LMI for Recurrent Networks (#997)

{
  "worked": true,
  "confidence": 9,
  "verdict": "Built a readable delay-margin MVP for a delayed scalar recurrent dynamics model using an Euler companion-state realization and the exact quadratic discrete Lyapunov LMI MᵀPM−P<0. The mechanism manifested: observed certified boundaries were 94.7–98.7% of analytic continuous-time predictions, alpha·h remained approximately constant (1.16–1.19), and below/above the predicted boundary the measured log amplitudes were −0.107/+0.064 per time unit. This is a strong toy verification, not evidence yet for a trained nonlinear RNN win.",
  "metrics": {
    "baseline": "At h=1.451 (> beta=2 predicted margin 1.209), unconstrained beta=2 had max |x|=7.59 and final |x|=0.895; it was LMI-infeasible with rho=1.00128.",
    "idea": "At the same delay, certificate-gated beta=1.25 had max |x|=1.00 and final |x|=0.00384. Across beta=1.1–3.0, observed LMI boundaries were 5.84, 3.28, 2.02, 1.18, 0.64 versus predictions 5.918, 3.331, 2.058, 1.209, 0.676."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 delay_margin_experiment.py",
  "files": [
    "delay_margin_experiment.py",
    "results.json"
  ],
  "limitations": "Did not implement the paper-specific continuous-time free-weighting-matrix Xi or solve a general semidefinite program. Did not train or benchmark a GRU/RNN on delayed-copy or sequential-MNIST, measure FLOPs/speed, estimate Jacobians from minibatches, test nonlinear residual ultimate boundedness, or compare against spectral normalization."
}