{
  "seed": 2771,
  "model": "dx/dt = x - x^3 + u (desired positive equilibrium)",
  "predictions": {
    "saddle_node": {
      "analytic_uc": 0.3849001794597505,
      "numeric_last_three_root_u": 0.3849,
      "absolute_error": 1.7945975050448482e-07
    },
    "local_vs_global": "desired Jacobian remains negative throughout the three-root bistable interval; local stability does not imply global recovery",
    "basin_transition": "at u=0, separatrix is x=0 and desired equilibrium is x=1; recovery probability is Phi(1/epsilon)"
  },
  "stability_sweep": [
    {
      "u": 0.0,
      "roots": [
        -1.0,
        0.0,
        1.0
      ],
      "desired_jacobian": -2.0,
      "locally_stable": true,
      "coexisting": true
    },
    {
      "u": 0.1,
      "roots": [
        -0.9456492739235919,
        -0.10103125788101083,
        1.0466805318046022
      ],
      "desired_jacobian": -2.2866204069762945,
      "locally_stable": true,
      "coexisting": true
    },
    {
      "u": 0.2,
      "roots": [
        -0.8788850662499734,
        -0.20914884844131656,
        1.0880339146912905
      ],
      "desired_jacobian": -2.5514533985553633,
      "locally_stable": true,
      "coexisting": true
    },
    {
      "u": 0.35,
      "roots": [
        -0.7140106835733218,
        -0.428896406402875,
        1.1429070899761962
      ],
      "desired_jacobian": -2.9187098489535708,
      "locally_stable": true,
      "coexisting": true
    },
    {
      "u": 0.38,
      "roots": [
        -0.6297529346989725,
        -0.5233111196073493,
        1.1530640543063222
      ],
      "desired_jacobian": -2.9886701399999995,
      "locally_stable": true,
      "coexisting": true
    }
  ],
  "basin_sweep": [
    {
      "epsilon": 0.1,
      "B_hat": 1.0,
      "B_predicted_gaussian": 1.0,
      "separatrix": 0.0,
      "reference": 1.0
    },
    {
      "epsilon": 0.3,
      "B_hat": 1.0,
      "B_predicted_gaussian": 0.9995709396668031,
      "separatrix": 0.0,
      "reference": 1.0
    },
    {
      "epsilon": 0.5,
      "B_hat": 0.97875,
      "B_predicted_gaussian": 0.9772498680518208,
      "separatrix": 0.0,
      "reference": 1.0
    },
    {
      "epsilon": 0.7,
      "B_hat": 0.9215,
      "B_predicted_gaussian": 0.9234362744901652,
      "separatrix": 0.0,
      "reference": 1.0
    },
    {
      "epsilon": 0.9,
      "B_hat": 0.8675,
      "B_predicted_gaussian": 0.8667397370974945,
      "separatrix": 0.0,
      "reference": 1.0
    },
    {
      "epsilon": 1.1,
      "B_hat": 0.83075,
      "B_predicted_gaussian": 0.818348929556551,
      "separatrix": 0.0,
      "reference": 1.0
    }
  ],
  "finite_horizon_comparison": {
    "threshold_only": {
      "failure_rate": 0.14333333333333334,
      "release_rate": 1.0
    },
    "basin_guard": {
      "failure_rate": 0.2,
      "release_rate": 0.0
    }
  }
}