1{
  2  "seed": 2771,
  3  "model": "dx/dt = x - x^3 + u (desired positive equilibrium)",
  4  "predictions": {
  5    "saddle_node": {
  6      "analytic_uc": 0.3849001794597505,
  7      "numeric_last_three_root_u": 0.3849,
  8      "absolute_error": 1.7945975050448482e-07
  9    },
 10    "local_vs_global": "desired Jacobian remains negative throughout the three-root bistable interval; local stability does not imply global recovery",
 11    "basin_transition": "at u=0, separatrix is x=0 and desired equilibrium is x=1; recovery probability is Phi(1/epsilon)"
 12  },
 13  "stability_sweep": [
 14    {
 15      "u": 0.0,
 16      "roots": [
 17        -1.0,
 18        0.0,
 19        1.0
 20      ],
 21      "desired_jacobian": -2.0,
 22      "locally_stable": true,
 23      "coexisting": true
 24    },
 25    {
 26      "u": 0.1,
 27      "roots": [
 28        -0.9456492739235919,
 29        -0.10103125788101083,
 30        1.0466805318046022
 31      ],
 32      "desired_jacobian": -2.2866204069762945,
 33      "locally_stable": true,
 34      "coexisting": true
 35    },
 36    {
 37      "u": 0.2,
 38      "roots": [
 39        -0.8788850662499734,
 40        -0.20914884844131656,
 41        1.0880339146912905
 42      ],
 43      "desired_jacobian": -2.5514533985553633,
 44      "locally_stable": true,
 45      "coexisting": true
 46    },
 47    {
 48      "u": 0.35,
 49      "roots": [
 50        -0.7140106835733218,
 51        -0.428896406402875,
 52        1.1429070899761962
 53      ],
 54      "desired_jacobian": -2.9187098489535708,
 55      "locally_stable": true,
 56      "coexisting": true
 57    },
 58    {
 59      "u": 0.38,
 60      "roots": [
 61        -0.6297529346989725,
 62        -0.5233111196073493,
 63        1.1530640543063222
 64      ],
 65      "desired_jacobian": -2.9886701399999995,
 66      "locally_stable": true,
 67      "coexisting": true
 68    }
 69  ],
 70  "basin_sweep": [
 71    {
 72      "epsilon": 0.1,
 73      "B_hat": 1.0,
 74      "B_predicted_gaussian": 1.0,
 75      "separatrix": 0.0,
 76      "reference": 1.0
 77    },
 78    {
 79      "epsilon": 0.3,
 80      "B_hat": 1.0,
 81      "B_predicted_gaussian": 0.9995709396668031,
 82      "separatrix": 0.0,
 83      "reference": 1.0
 84    },
 85    {
 86      "epsilon": 0.5,
 87      "B_hat": 0.97875,
 88      "B_predicted_gaussian": 0.9772498680518208,
 89      "separatrix": 0.0,
 90      "reference": 1.0
 91    },
 92    {
 93      "epsilon": 0.7,
 94      "B_hat": 0.9215,
 95      "B_predicted_gaussian": 0.9234362744901652,
 96      "separatrix": 0.0,
 97      "reference": 1.0
 98    },
 99    {
100      "epsilon": 0.9,
101      "B_hat": 0.8675,
102      "B_predicted_gaussian": 0.8667397370974945,
103      "separatrix": 0.0,
104      "reference": 1.0
105    },
106    {
107      "epsilon": 1.1,
108      "B_hat": 0.83075,
109      "B_predicted_gaussian": 0.818348929556551,
110      "separatrix": 0.0,
111      "reference": 1.0
112    }
113  ],
114  "finite_horizon_comparison": {
115    "threshold_only": {
116      "failure_rate": 0.14333333333333334,
117      "release_rate": 1.0
118    },
119    "basin_guard": {
120      "failure_rate": 0.2,
121      "release_rate": 0.0
122    }
123  }
124}