{
  "seed": 1120,
  "model": "h[t+1]=r R(omega) h[t]+sigma Normal(0,I), y=h[0]",
  "predictions": {
    "decay_time": "tau=-1/log(r); fitted resonance radius predicts correlation decay",
    "frequency": "PSD peak and fitted omega equal injected omega",
    "independence": "additive noise changes variance, not J-based Lambda or resonance"
  },
  "radius_sweep": [
    {
      "r": 0.7,
      "fitted_r": 0.7034267653385271,
      "tau_pred": 2.8036732520571284,
      "tau_fit": 2.842592775655428,
      "lyapunov": -0.35667494393873245,
      "rel_tau_error": 0.013881618897545741
    },
    {
      "r": 0.8,
      "fitted_r": 0.8000406392805042,
      "tau_pred": 4.481420117724551,
      "tau_fit": 4.482440528857972,
      "lyapunov": -0.22314355131420976,
      "rel_tau_error": 0.00022769816411214353
    },
    {
      "r": 0.9,
      "fitted_r": 0.8999502249921175,
      "tau_pred": 9.491221581029905,
      "tau_fit": 9.486241950194362,
      "lyapunov": -0.10536051565782632,
      "rel_tau_error": 0.000524656472618374
    },
    {
      "r": 0.95,
      "fitted_r": 0.9513785486439428,
      "tau_pred": 19.495725746223673,
      "tau_fit": 20.06290024762786,
      "lyapunov": -0.05129329438755065,
      "rel_tau_error": 0.029092248669637097
    },
    {
      "r": 0.98,
      "fitted_r": 0.9797565041189408,
      "tau_pred": 49.49831645250911,
      "tau_fit": 48.89687817730976,
      "lyapunov": -0.020202707317519462,
      "rel_tau_error": 0.012150681443406185
    }
  ],
  "frequency_sweep": [
    {
      "omega": 0.25,
      "fitted_omega": 0.24863147848964415,
      "psd_peak": 0.26030339129744,
      "predicted": 0.25,
      "peak_abs_error": 0.010303391297440012,
      "relative_error": 0.04121356518976005
    },
    {
      "omega": 0.42,
      "fitted_omega": 0.4185664231447004,
      "psd_peak": 0.4146902302738527,
      "predicted": 0.42,
      "peak_abs_error": 0.005309769726147273,
      "relative_error": 0.012642308871779223
    },
    {
      "omega": 0.7,
      "fitted_omega": 0.6997844235370086,
      "psd_peak": 0.7126927334143702,
      "predicted": 0.7,
      "peak_abs_error": 0.012692733414370272,
      "relative_error": 0.018132476306243248
    },
    {
      "omega": 1.0,
      "fitted_omega": 0.999159863137488,
      "psd_peak": 1.0035144533466827,
      "predicted": 1.0,
      "peak_abs_error": 0.0035144533466826555,
      "relative_error": 0.0035144533466826555
    }
  ],
  "noise_sweep": [
    {
      "sigma": 0.03,
      "fitted_r": 0.9314177422652783,
      "fitted_omega": 0.4193830018943419,
      "lyapunov": -0.07257069283483542,
      "variance": 0.006688245206206876
    },
    {
      "sigma": 0.12,
      "fitted_r": 0.9299692025798508,
      "fitted_omega": 0.4193736326910304,
      "lyapunov": -0.07257069283483542,
      "variance": 0.10755494007243742
    },
    {
      "sigma": 0.4,
      "fitted_r": 0.9303895923523557,
      "fitted_omega": 0.4222108423811429,
      "lyapunov": -0.07257069283483542,
      "variance": 1.1937186348619224
    },
    {
      "sigma": 0.8,
      "fitted_r": 0.9298517654805027,
      "fitted_omega": 0.4196132638016788,
      "lyapunov": -0.07257069283483542,
      "variance": 4.7345430222524385
    }
  ],
  "control": {
    "uncontrolled": {
      "fitted_r": 0.9752652124757705,
      "fitted_omega": 0.4188032303923567,
      "decay_time": 39.926802707809806,
      "peak_frequency": 0.4194026192542374,
      "peak_power": 25.511866235218225,
      "long_lag_corr": 0.06225150327776391,
      "lyapunov": -0.02531780798428996
    },
    "resonance_control": {
      "fitted_r": 0.8996346421972676,
      "fitted_omega": 0.4183753804168158,
      "decay_time": 9.454784983703435,
      "peak_frequency": 0.4194026192542374,
      "peak_power": 1.7077026060796374,
      "long_lag_corr": 0.004897474469038448,
      "lyapunov": -0.10536051565782632
    },
    "target_radius": 0.9
  },
  "quantitative_checks": {
    "decay_time_max_relative_error": 0.029092248669637097,
    "decay_time_pass_under_5pct": true,
    "frequency_max_relative_error": 0.04121356518976005,
    "frequency_pass_under_10pct": true,
    "noise_fitted_radius_span": 0.0015659767847755468,
    "noise_fitted_frequency_span": 0.00283720969011253,
    "noise_invariance_pass_under_0.01": true,
    "baseline_peak_power": 25.511866235218225,
    "idea_peak_power": 1.7077026060796374,
    "peak_power_reduction_factor": 14.939291036034467,
    "baseline_long_lag_abs_corr": 0.06225150327776391,
    "idea_long_lag_abs_corr": 0.004897474469038448
  }
}