{
  "seed": 1418,
  "linear_contraction_sweep": {
    "predicted_stability_boundary_g2": 2.0,
    "observed_first_grid_stable_g2": 2.0500000000000003,
    "grid_step": 0.05,
    "projected_H": 0.5,
    "checks": {
      "1.5": {
        "spectral_abscissa": 0.28077640640441515,
        "log_norm_slope": 0.28506515724491827,
        "norm_t0": 1.4142135623730951,
        "norm_t8": 13.920981013765749
      },
      "3.0": {
        "spectral_abscissa": -0.7499999999999999,
        "log_norm_slope": -0.7524656038095815,
        "norm_t0": 1.4142135623730951,
        "norm_t8": 0.012201718698289765
      }
    }
  },
  "gain_sweep": [
    {
      "g2": 0.0,
      "full_abscissa": 0.8507810593582121,
      "projected_H": 0.5
    },
    {
      "g2": 0.25,
      "full_abscissa": 0.770690632574555,
      "projected_H": 0.5
    },
    {
      "g2": 0.5,
      "full_abscissa": 0.6861406616345072,
      "projected_H": 0.5
    },
    {
      "g2": 0.75,
      "full_abscissa": 0.596291201783626,
      "projected_H": 0.5
    },
    {
      "g2": 1.0,
      "full_abscissa": 0.5,
      "projected_H": 0.5
    },
    {
      "g2": 1.25,
      "full_abscissa": 0.39564392373896,
      "projected_H": 0.5
    },
    {
      "g2": 1.5,
      "full_abscissa": 0.28077640640441515,
      "projected_H": 0.5
    },
    {
      "g2": 1.75,
      "full_abscissa": 0.15138781886599734,
      "projected_H": 0.5
    },
    {
      "g2": 2.0,
      "full_abscissa": 0.0,
      "projected_H": 0.5
    },
    {
      "g2": 2.25,
      "full_abscissa": -0.19098300562505255,
      "projected_H": 0.5
    },
    {
      "g2": 2.5,
      "full_abscissa": -0.4999999999999999,
      "projected_H": 0.5
    },
    {
      "g2": 2.75,
      "full_abscissa": -0.7499999999999998,
      "projected_H": 0.5
    },
    {
      "g2": 3.0,
      "full_abscissa": -0.7499999999999999,
      "projected_H": 0.5
    },
    {
      "g2": 3.25,
      "full_abscissa": -0.7499999999999999,
      "projected_H": 0.5
    },
    {
      "g2": 3.5,
      "full_abscissa": -0.7500000000000001,
      "projected_H": 0.5
    },
    {
      "g2": 3.75,
      "full_abscissa": -0.75,
      "projected_H": 0.5
    },
    {
      "g2": 4.0,
      "full_abscissa": -0.75,
      "projected_H": 0.5
    }
  ],
  "euler_stability_sweep": {
    "g2": 3.0,
    "eigenvalues": [
      [
        -0.7499999999999999,
        0.6614378277661476
      ],
      [
        -0.7499999999999999,
        -0.6614378277661476
      ]
    ],
    "predicted_dt_max": 1.5000000000000002,
    "observed_dt_max_grid": 1.4989999999999999,
    "dt_grid": 0.001
  },
  "lorenz_partial_observation": {
    "open_loop": {
      "initial_error": 20.149441679609886,
      "final_error": 30.97985872769628,
      "median_last_20pct": 14.828022853040249
    },
    "nudging": {
      "initial_error": 20.149441679609886,
      "final_error": 0.12642521488225347,
      "median_last_20pct": 0.14382116971552625
    }
  },
  "interpretation": {
    "prediction_1": "Full 2D observer stability begins at g2=2 from det(A-GC)>0; observed grid boundary should agree.",
    "prediction_2": "At stable g2=3, Euler stability has a finite dt boundary predicted by -2 Re(lambda)/|lambda|^2; observed sweep should agree.",
    "prediction_gain": "Projected H remains +0.5 across all gains, while full abscissa crosses zero at g2=2; this falsifies H-only sufficiency here.",
    "prediction_3": "The proposed H on ker(C) remains +0.5 for every gain, despite full observer becoming stable; therefore the stated projected certificate is not sufficient in this coupled example."
  }
}