{
  "device": "cpu",
  "predictions": {
    "P1": "nonnegative W,w => Jensen violation <= numerical tolerance (predicted 0)",
    "P2": "negative W entries create positive Jensen violations; violation rate rises with negative fraction",
    "P3": "positive output coefficients make ReLU epigraph LP exact (predicted objective gap 0)"
  },
  "convexity_sweep": [
    {
      "negative_fraction": 0.0,
      "max_jensen_violation": 2.842170943040401e-14,
      "violation_rate": 0.0,
      "mean_positive_violation": 1.1421049291489757e-16
    },
    {
      "negative_fraction": 0.05,
      "max_jensen_violation": 4.263256414560601e-14,
      "violation_rate": 0.0,
      "mean_positive_violation": 4.902300787534842e-16
    },
    {
      "negative_fraction": 0.1,
      "max_jensen_violation": 7.105427357601002e-15,
      "violation_rate": 0.0,
      "mean_positive_violation": 4.1011840914058814e-17
    },
    {
      "negative_fraction": 0.25,
      "max_jensen_violation": 1.0658141036401503e-14,
      "violation_rate": 0.0,
      "mean_positive_violation": 4.912366809624776e-17
    },
    {
      "negative_fraction": 0.5,
      "max_jensen_violation": 1.4250197781545362,
      "violation_rate": 0.22593333333333335,
      "mean_positive_violation": 0.03872182267024236
    },
    {
      "negative_fraction": 1.0,
      "max_jensen_violation": 1.5431175754627462,
      "violation_rate": 0.2742,
      "mean_positive_violation": 0.07106292150874202
    }
  ],
  "controlled_sign_sweep": [
    {
      "gamma": -2.0,
      "predicted_max_violation": 0.5,
      "observed_max_violation": 0.5
    },
    {
      "gamma": -1.0,
      "predicted_max_violation": 0.5,
      "observed_max_violation": 0.5
    },
    {
      "gamma": -0.5,
      "predicted_max_violation": 0.25,
      "observed_max_violation": 0.25
    },
    {
      "gamma": -0.1,
      "predicted_max_violation": 0.05,
      "observed_max_violation": 0.050000000000000044
    },
    {
      "gamma": 0.0,
      "predicted_max_violation": 0.0,
      "observed_max_violation": 0.0
    },
    {
      "gamma": 0.1,
      "predicted_max_violation": 0.0,
      "observed_max_violation": 2.220446049250313e-16
    },
    {
      "gamma": 0.5,
      "predicted_max_violation": 0.0,
      "observed_max_violation": 4.440892098500626e-16
    },
    {
      "gamma": 1.0,
      "predicted_max_violation": 0.0,
      "observed_max_violation": 2.220446049250313e-16
    },
    {
      "gamma": 2.0,
      "predicted_max_violation": 0.0,
      "observed_max_violation": 4.440892098500626e-16
    }
  ],
  "epigraph_check": {
    "one_layer_max_abs_gap": 0.0,
    "two_layer_max_abs_gap": 1.7763568394002505e-15,
    "predicted_gap": 0.0
  },
  "fit_comparison": {
    "icnn": {
      "train_mse": 1.0684418678283691,
      "validation_mse": 1.2404754161834717,
      "steps": 1200
    },
    "relu_mlp": {
      "train_mse": 0.005368055310100317,
      "validation_mse": 0.007096992339938879,
      "steps": 1200
    },
    "metric": "validation MSE (lower is better)"
  }
}