{
  "seed": 3140,
  "alpha_exact_threshold": 0.73,
  "representation_checks": {
    "h": {
      "D_at_rho_1": 0.73,
      "raw_mean_r": 1.4600000000000004
    },
    "2h": {
      "D_at_rho_1": 0.73,
      "raw_mean_r": 2.920000000000001
    },
    "h+h^3": {
      "D_at_rho_1": 0.73,
      "raw_mean_r": 1.4600000000000004
    },
    "log(1+h)": {
      "D_at_rho_1": 0.73,
      "raw_mean_r": 1.4600000000000004
    },
    "h^3 (invalid)": {
      "D_at_rho_1": Infinity,
      "raw_mean_r": 1.9589033626083181e-32
    }
  },
  "transition": [
    {
      "rho": 0.5,
      "D": 1.46,
      "min_boundary_derivative": -0.45999999999999996,
      "feasible_sampled": false
    },
    {
      "rho": 0.65,
      "D": 1.123076923076923,
      "min_boundary_derivative": -0.15999999999999992,
      "feasible_sampled": false
    },
    {
      "rho": 0.72,
      "D": 1.0138888888888888,
      "min_boundary_derivative": -0.020000000000000018,
      "feasible_sampled": false
    },
    {
      "rho": 0.73,
      "D": 1.0,
      "min_boundary_derivative": 0.0,
      "feasible_sampled": true
    },
    {
      "rho": 0.74,
      "D": 0.9864864864864865,
      "min_boundary_derivative": 0.020000000000000018,
      "feasible_sampled": true
    },
    {
      "rho": 0.85,
      "D": 0.8588235294117647,
      "min_boundary_derivative": 0.24,
      "feasible_sampled": true
    },
    {
      "rho": 1.0,
      "D": 0.73,
      "min_boundary_derivative": 0.54,
      "feasible_sampled": true
    }
  ],
  "softsup_tau_0.02_rho_1": 0.6647435462271849,
  "max_boundary_gradient_finite_difference_error": 1.3789791530882667e-10,
  "valid_invariance_relative_spread": 0.0,
  "invalid_h3_D": Infinity,
  "notes": "For h^3, q_prime(0)=0, so the defining-function regularity assumption fails and the boundary gradient vanishes."
}