{
  "device": "cuda",
  "seed": 1445,
  "math_check": {
    "max_radial_identity_error": 6.938893903907228e-16,
    "speed_decay_sweep": [
      {
        "kv": 0.5,
        "predicted_rate": 0.5,
        "observed_rate": 0.5,
        "max_identity_error": 1.1102230246251565e-16
      },
      {
        "kv": 1.0,
        "predicted_rate": 1.0,
        "observed_rate": 1.0,
        "max_identity_error": 2.220446049250313e-16
      },
      {
        "kv": 2.0,
        "predicted_rate": 2.0,
        "observed_rate": 2.0,
        "max_identity_error": 0.0
      },
      {
        "kv": 3.0,
        "predicted_rate": 3.0,
        "observed_rate": 3.0,
        "max_identity_error": 2.220446049250313e-16
      }
    ],
    "steering_decay_sweep": [
      {
        "v": 0.01,
        "predicted_rate": 0.19999999999999998,
        "observed_rate": 0.20000000000000043,
        "relative_error": 2.220446049250313e-15
      },
      {
        "v": 0.05,
        "predicted_rate": 0.6,
        "observed_rate": 0.5999999999999995,
        "relative_error": 7.401486830834377e-16
      },
      {
        "v": 0.1,
        "predicted_rate": 0.7999999999999998,
        "observed_rate": 0.8000000000000008,
        "relative_error": 1.2490009027033013e-15
      },
      {
        "v": 0.2,
        "predicted_rate": 0.96,
        "observed_rate": 0.9600000000000009,
        "relative_error": 9.251858538542972e-16
      },
      {
        "v": 0.5,
        "predicted_rate": 1.0909090909090908,
        "observed_rate": 1.0909090909090902,
        "relative_error": 6.106226635438361e-16
      },
      {
        "v": 1.0,
        "predicted_rate": 1.1428571428571428,
        "observed_rate": 1.1428571428571421,
        "relative_error": 5.828670879282072e-16
      },
      {
        "v": 2.0,
        "predicted_rate": 1.1707317073170733,
        "observed_rate": 1.1707317073170733,
        "relative_error": 0.0
      }
    ],
    "epsilon_transition_sweep": [
      {
        "r": 0.01,
        "predicted_abs_geometric_residual": 30.683234470668992,
        "observed_abs_gamma_residual": 30.68323447066898,
        "relative_error": 3.4736041425456412e-16
      },
      {
        "r": 0.025,
        "predicted_abs_geometric_residual": 7.670808617667248,
        "observed_abs_gamma_residual": 7.670808617667237,
        "relative_error": 1.3894416570182565e-15
      },
      {
        "r": 0.04,
        "predicted_abs_geometric_residual": 1.917702154416812,
        "observed_abs_gamma_residual": 1.9177021544168014,
        "relative_error": 5.557766628073026e-15
      },
      {
        "r": 0.05,
        "predicted_abs_geometric_residual": 0.0,
        "observed_abs_gamma_residual": 1.0547118733938987e-14,
        "relative_error": 1.0547118733938987e-06
      },
      {
        "r": 0.075,
        "predicted_abs_geometric_residual": 0.0,
        "observed_abs_gamma_residual": 1.1102230246251565e-16,
        "relative_error": 1.1102230246251565e-08
      },
      {
        "r": 0.1,
        "predicted_abs_geometric_residual": 0.0,
        "observed_abs_gamma_residual": 1.1102230246251565e-16,
        "relative_error": 1.1102230246251565e-08
      },
      {
        "r": 0.2,
        "predicted_abs_geometric_residual": 0.0,
        "observed_abs_gamma_residual": 9.992007221626409e-16,
        "relative_error": 9.992007221626409e-08
      },
      {
        "r": 0.5,
        "predicted_abs_geometric_residual": 0.0,
        "observed_abs_gamma_residual": 1.1102230246251565e-16,
        "relative_error": 1.1102230246251565e-08
      },
      {
        "r": 1.0,
        "predicted_abs_geometric_residual": 0.0,
        "observed_abs_gamma_residual": 1.1102230246251565e-16,
        "relative_error": 1.1102230246251565e-08
      }
    ]
  },
  "policy_comparison": {
    "unconstrained_polar_residual": {
      "task_mse": 0.0007122126407921314,
      "positive_drift_fraction": 0.130859375,
      "mean_drift_plus_lambdaV": -0.1348373144865036,
      "mean_V": 1.8700960874557495
    },
    "lyapunov_residual": {
      "task_mse": 0.06113187596201897,
      "positive_drift_fraction": 0.10888671875,
      "mean_drift_plus_lambdaV": -0.20551252365112305,
      "mean_V": 1.885925531387329
    }
  },
  "interpretation": {
    "speed_prediction": "e_dot=-kv*e exactly",
    "steering_prediction": "-gamma_dot/gamma=kg*v/(abs(v)+epsilon) for positive v",
    "epsilon_prediction": "geometric gamma residual is exactly zero for r>=epsilon and scales as 1/r-1/epsilon below epsilon"
  }
}