{
  "math_verification": {
    "prediction_1_secant_exactness": "s.T ytilde / abs(s.T y) = 1 for every rho",
    "prediction_2_correction_scaling": "||ytilde-y||/||y|| = 2 rho for prescribed c=-rho||s||||y||",
    "prediction_3_positive_definiteness": "BFGS corrected update has strictly positive eigenvalues when B is SPD",
    "sweep": [
      {
        "rho": 0.001,
        "observed_secant_over_abs_c": 0.9999999999999867,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 0.0020000000000001397,
        "predicted_delta_ratio": 0.002
      },
      {
        "rho": 0.01,
        "observed_secant_over_abs_c": 1.000000000000001,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 0.019999999999999907,
        "predicted_delta_ratio": 0.02
      },
      {
        "rho": 0.05,
        "observed_secant_over_abs_c": 1.0000000000000004,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 0.09999999999999987,
        "predicted_delta_ratio": 0.1
      },
      {
        "rho": 0.1,
        "observed_secant_over_abs_c": 1.0000000000000002,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 0.20000000000000018,
        "predicted_delta_ratio": 0.2
      },
      {
        "rho": 0.25,
        "observed_secant_over_abs_c": 0.9999999999999999,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 0.5000000000000002,
        "predicted_delta_ratio": 0.5
      },
      {
        "rho": 0.5,
        "observed_secant_over_abs_c": 1.0,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 1.0000000000000002,
        "predicted_delta_ratio": 1.0
      },
      {
        "rho": 0.9,
        "observed_secant_over_abs_c": 0.9999999999999999,
        "predicted_secant_over_abs_c": 1.0,
        "observed_delta_ratio": 1.7999999999999998,
        "predicted_delta_ratio": 1.8
      }
    ],
    "spd_sweep": [
      {
        "dimension": 3,
        "minimum_observed_eigenvalue": 0.03528108020571712,
        "predicted_lower_bound": 0.0
      },
      {
        "dimension": 8,
        "minimum_observed_eigenvalue": 0.00927656923766997,
        "predicted_lower_bound": 0.0
      },
      {
        "dimension": 20,
        "minimum_observed_eigenvalue": 0.0010288274956301389,
        "predicted_lower_bound": 0.0
      }
    ]
  },
  "mini_experiment": {
    "baseline_ordinary_reject": {
      "initial_loss": -0.05734205502046099,
      "final_loss": -0.625,
      "best_loss": -0.625,
      "negative_pairs": 1,
      "corrected_pairs": 0,
      "backtracking_halvings": 164,
      "steps": 180,
      "finite": true
    },
    "idea_euclidean_correction": {
      "initial_loss": -0.05734205502046099,
      "final_loss": -0.625,
      "best_loss": -0.625,
      "negative_pairs": 1,
      "corrected_pairs": 1,
      "backtracking_halvings": 1,
      "steps": 180,
      "finite": true
    }
  }
}