1{
  2  "seed": 1418,
  3  "linear_contraction_sweep": {
  4    "predicted_stability_boundary_g2": 2.0,
  5    "observed_first_grid_stable_g2": 2.0500000000000003,
  6    "grid_step": 0.05,
  7    "projected_H": 0.5,
  8    "checks": {
  9      "1.5": {
 10        "spectral_abscissa": 0.28077640640441515,
 11        "log_norm_slope": 0.28506515724491827,
 12        "norm_t0": 1.4142135623730951,
 13        "norm_t8": 13.920981013765749
 14      },
 15      "3.0": {
 16        "spectral_abscissa": -0.7499999999999999,
 17        "log_norm_slope": -0.7524656038095815,
 18        "norm_t0": 1.4142135623730951,
 19        "norm_t8": 0.012201718698289765
 20      }
 21    }
 22  },
 23  "gain_sweep": [
 24    {
 25      "g2": 0.0,
 26      "full_abscissa": 0.8507810593582121,
 27      "projected_H": 0.5
 28    },
 29    {
 30      "g2": 0.25,
 31      "full_abscissa": 0.770690632574555,
 32      "projected_H": 0.5
 33    },
 34    {
 35      "g2": 0.5,
 36      "full_abscissa": 0.6861406616345072,
 37      "projected_H": 0.5
 38    },
 39    {
 40      "g2": 0.75,
 41      "full_abscissa": 0.596291201783626,
 42      "projected_H": 0.5
 43    },
 44    {
 45      "g2": 1.0,
 46      "full_abscissa": 0.5,
 47      "projected_H": 0.5
 48    },
 49    {
 50      "g2": 1.25,
 51      "full_abscissa": 0.39564392373896,
 52      "projected_H": 0.5
 53    },
 54    {
 55      "g2": 1.5,
 56      "full_abscissa": 0.28077640640441515,
 57      "projected_H": 0.5
 58    },
 59    {
 60      "g2": 1.75,
 61      "full_abscissa": 0.15138781886599734,
 62      "projected_H": 0.5
 63    },
 64    {
 65      "g2": 2.0,
 66      "full_abscissa": 0.0,
 67      "projected_H": 0.5
 68    },
 69    {
 70      "g2": 2.25,
 71      "full_abscissa": -0.19098300562505255,
 72      "projected_H": 0.5
 73    },
 74    {
 75      "g2": 2.5,
 76      "full_abscissa": -0.4999999999999999,
 77      "projected_H": 0.5
 78    },
 79    {
 80      "g2": 2.75,
 81      "full_abscissa": -0.7499999999999998,
 82      "projected_H": 0.5
 83    },
 84    {
 85      "g2": 3.0,
 86      "full_abscissa": -0.7499999999999999,
 87      "projected_H": 0.5
 88    },
 89    {
 90      "g2": 3.25,
 91      "full_abscissa": -0.7499999999999999,
 92      "projected_H": 0.5
 93    },
 94    {
 95      "g2": 3.5,
 96      "full_abscissa": -0.7500000000000001,
 97      "projected_H": 0.5
 98    },
 99    {
100      "g2": 3.75,
101      "full_abscissa": -0.75,
102      "projected_H": 0.5
103    },
104    {
105      "g2": 4.0,
106      "full_abscissa": -0.75,
107      "projected_H": 0.5
108    }
109  ],
110  "euler_stability_sweep": {
111    "g2": 3.0,
112    "eigenvalues": [
113      [
114        -0.7499999999999999,
115        0.6614378277661476
116      ],
117      [
118        -0.7499999999999999,
119        -0.6614378277661476
120      ]
121    ],
122    "predicted_dt_max": 1.5000000000000002,
123    "observed_dt_max_grid": 1.4989999999999999,
124    "dt_grid": 0.001
125  },
126  "lorenz_partial_observation": {
127    "open_loop": {
128      "initial_error": 20.149441679609886,
129      "final_error": 30.97985872769628,
130      "median_last_20pct": 14.828022853040249
131    },
132    "nudging": {
133      "initial_error": 20.149441679609886,
134      "final_error": 0.12642521488225347,
135      "median_last_20pct": 0.14382116971552625
136    }
137  },
138  "interpretation": {
139    "prediction_1": "Full 2D observer stability begins at g2=2 from det(A-GC)>0; observed grid boundary should agree.",
140    "prediction_2": "At stable g2=3, Euler stability has a finite dt boundary predicted by -2 Re(lambda)/|lambda|^2; observed sweep should agree.",
141    "prediction_gain": "Projected H remains +0.5 across all gains, while full abscissa crosses zero at g2=2; this falsifies H-only sufficiency here.",
142    "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."
143  }
144}