{
  "seed": 7,
  "math_check": {
    "P": [
      [
        1.2500000000000002,
        0.25
      ],
      [
        0.25000000000000006,
        0.37500000000000006
      ]
    ],
    "identity_error": 1.1102230246251565e-16,
    "gamma": 0.002734749554788495,
    "max_backup_u_on_certificate": 0.17404742623406258,
    "observed_min_neg_vdot_over_V": 0.7596534626874615,
    "predicted_rate_bound": 0.7596526541079154,
    "certificate_saturation_free": true
  },
  "prediction_1_dissipation": "-Vdot/V >= 1/lambda_max(P) on unsaturated certificate",
  "prediction_2_horizon": "backward reachable volume is nondecreasing with T",
  "prediction_3_scaling": "quadratic certificate area scales as terminal_radius^2",
  "certificate_scaling_sweep": [
    {
      "terminal_radius": 0.03,
      "certified_area": 0.0033698514343115014,
      "area_over_base": 0.25,
      "predicted_ratio": 0.25
    },
    {
      "terminal_radius": 0.06,
      "certified_area": 0.013479405737246005,
      "area_over_base": 1.0,
      "predicted_ratio": 1.0
    },
    {
      "terminal_radius": 0.12,
      "certified_area": 0.05391762294898402,
      "area_over_base": 4.0,
      "predicted_ratio": 4.0
    },
    {
      "terminal_radius": 0.18,
      "certified_area": 0.12131465163521406,
      "area_over_base": 9.0,
      "predicted_ratio": 9.0
    }
  ],
  "reachability_sweep": [
    {
      "T": 0.5,
      "volume": 0.025200000000000007,
      "fraction": 0.0018812147272238647
    },
    {
      "T": 1.0,
      "volume": 0.08280000000000001,
      "fraction": 0.006181134103735555
    },
    {
      "T": 2.0,
      "volume": 0.6228000000000001,
      "fraction": 0.046492878258532655
    },
    {
      "T": 3.0,
      "volume": 3.387600000000001,
      "fraction": 0.2528890083310938
    },
    {
      "T": 4.0,
      "volume": 8.341200000000002,
      "fraction": 0.6226820747110992
    }
  ],
  "input_saturation_sweep": [
    {
      "kp": 0.5,
      "saturated_fraction": 0.7165775401069518
    },
    {
      "kp": 1.0,
      "saturated_fraction": 0.7486631016042781
    },
    {
      "kp": 2.0,
      "saturated_fraction": 0.6737967914438503
    },
    {
      "kp": 4.0,
      "saturated_fraction": 0.9090909090909091
    },
    {
      "kp": 8.0,
      "saturated_fraction": 0.9090909090909091
    }
  ],
  "certified_rollout_audit": {
    "n": 500,
    "safe_rate": 1.0,
    "terminal_rate": 1.0,
    "max_violation": 0.0,
    "horizon": 4.0
  },
  "policy_comparison": {
    "n": 1000,
    "baseline_violation_rate": 0.962,
    "shield_violation_rate": 0.228,
    "baseline_mean_violation": 14.94174058032316,
    "shield_mean_violation": 0.08638416745996397,
    "shield_mean_switches": 628.568
  }
}