{
  "seed": 1411,
  "derivative_bound_check": {
    "predicted_max": 0.5873015873015873,
    "observed_max": 0.5873012716278044,
    "relative_error": 5.374986033014429e-07,
    "bound_location": 0.0,
    "analytic_numeric_max_gap": 3.1567378289132364e-07
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  "bias_scaling_prediction": {
    "predicted_slope": 0.9576460059232068,
    "observed_slope": 0.9562668702882926,
    "slope_ratio_observed_over_predicted": 0.9985598690681274,
    "rows": [
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        "alpha": 0.0,
        "predicted_linear_bias": 0.0,
        "observed_baseline_b_bias": 0.00013437299130325542,
        "joint_train_mse": 9.944095552998701e-05,
        "baseline_cross_regime_mse": 0.00010193571553171425,
        "joint_cross_regime_mse": 0.00010189734634527286,
        "fitted_alpha": 0.0001341819738962893,
        "fitted_tau": 0.23011589497233
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        "joint_train_mse": 9.580818687379242e-05,
        "baseline_cross_regime_mse": 0.00043928052870968753,
        "joint_cross_regime_mse": 0.00010018082671487575,
        "fitted_alpha": 0.05137334370145234,
        "fitted_tau": 0.7992520245044272
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      {
        "alpha": 0.1,
        "predicted_linear_bias": 0.09554581666349103,
        "observed_baseline_b_bias": 0.09560453428039284,
        "joint_train_mse": 9.742284768823036e-05,
        "baseline_cross_regime_mse": 0.0014248311197335368,
        "joint_cross_regime_mse": 9.843910234062992e-05,
        "fitted_alpha": 0.09742946903222409,
        "fitted_tau": 0.7837634760645263
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      {
        "alpha": 0.2,
        "predicted_linear_bias": 0.19164426773318896,
        "observed_baseline_b_bias": 0.1912787636023856,
        "joint_train_mse": 9.918944042284518e-05,
        "baseline_cross_regime_mse": 0.005747213710342346,
        "joint_cross_regime_mse": 0.00010099567845155088,
        "fitted_alpha": 0.2106558905749331,
        "fitted_tau": 0.8252081548622062
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      {
        "alpha": 0.4,
        "predicted_linear_bias": 0.38227219901095233,
        "observed_baseline_b_bias": 0.3823734017989071,
        "joint_train_mse": 9.843000240422399e-05,
        "baseline_cross_regime_mse": 0.02307951568052862,
        "joint_cross_regime_mse": 0.00010030937763751038,
        "fitted_alpha": 0.3909552665125492,
        "fitted_tau": 0.7908966400771289
      },
      {
        "alpha": 0.8,
        "predicted_linear_bias": 0.7658530171517917,
        "observed_baseline_b_bias": 0.7660247127607649,
        "joint_train_mse": 0.00010161823047893185,
        "baseline_cross_regime_mse": 0.09097344100691553,
        "joint_cross_regime_mse": 9.933993582472864e-05,
        "fitted_alpha": 0.7977660621050289,
        "fitted_tau": 0.8000909968069085
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    ]
  },
  "zero_disturbance_prediction": {
    "mean_baseline_mse": 9.856082351256858e-05,
    "mean_joint_mse": 9.90246763733528e-05,
    "joint_over_baseline": 1.0047062599951295,
    "rows": [
      {
        "baseline_mse": 9.32833170934401e-05,
        "joint_mse": 9.347766786106346e-05,
        "fitted_alpha": -0.0018359754442777662
      },
      {
        "baseline_mse": 0.00010427287559700058,
        "joint_mse": 0.00010521555921400823,
        "fitted_alpha": 0.0018337445141214646
      },
      {
        "baseline_mse": 9.666887066118742e-05,
        "joint_mse": 9.706605944360958e-05,
        "fitted_alpha": -0.0009042560167786406
      },
      {
        "baseline_mse": 0.00010147152010286741,
        "joint_mse": 0.00010236196998820569,
        "fitted_alpha": -0.003272748414415243
      },
      {
        "baseline_mse": 9.788786176946618e-05,
        "joint_mse": 9.834246801682991e-05,
        "fitted_alpha": 0.0025736185237446737
      },
      {
        "baseline_mse": 9.986842096483505e-05,
        "joint_mse": 0.00010024066899184692,
        "fitted_alpha": 0.0011866098219361155
      },
      {
        "baseline_mse": 9.70123687942745e-05,
        "joint_mse": 9.737927636621525e-05,
        "fitted_alpha": 0.012784153980822872
      },
      {
        "baseline_mse": 9.802135311747739e-05,
        "joint_mse": 9.811374110504336e-05,
        "fitted_alpha": -0.0003468979252745945
      }
    ]
  },
  "interpretation": "Joint model should recover saturation and improve held-out command-regime error when alpha is nonzero; at alpha=0 no systematic benefit is expected."
}