{
  "seed": 1502,
  "structural": [
    {
      "design": "matched",
      "reachable": 10,
      "n": 10,
      "core_nonzero": 10,
      "matching": 10,
      "row_saturating": true
    },
    {
      "design": "unmatched",
      "reachable": 10,
      "n": 10,
      "core_nonzero": 10,
      "matching": 2,
      "row_saturating": false
    },
    {
      "design": "inaccessible",
      "reachable": 9,
      "n": 10,
      "core_nonzero": 9,
      "matching": 9,
      "row_saturating": false
    },
    {
      "design": "random",
      "reachable": 10,
      "n": 10,
      "core_nonzero": 125,
      "matching": 10,
      "row_saturating": true
    }
  ],
  "gramian_sweep": [
    {
      "design": "matched",
      "mean_ctrb_rank": 10.0,
      "mean_gramian_rank": 9.25,
      "min_gramian_rank": 7,
      "mean_lambda_min": 1.4744208487949544e-06,
      "median_lambda_min": 3.8187774199632105e-08
    },
    {
      "design": "unmatched",
      "mean_ctrb_rank": 2.0,
      "mean_gramian_rank": 2.0,
      "min_gramian_rank": 2,
      "mean_lambda_min": -7.711602072991294e-15,
      "median_lambda_min": -3.514073166814658e-15
    },
    {
      "design": "inaccessible",
      "mean_ctrb_rank": 9.0,
      "mean_gramian_rank": 5.25,
      "min_gramian_rank": 3,
      "mean_lambda_min": 0.0,
      "median_lambda_min": 0.0
    },
    {
      "design": "random",
      "mean_ctrb_rank": 9.0,
      "mean_gramian_rank": 8.75,
      "min_gramian_rank": 8,
      "mean_lambda_min": -2.5771304018052742e-12,
      "median_lambda_min": 9.90252208757925e-15
    }
  ],
  "matched_horizon_sweep": [
    {
      "T": 0.25,
      "lambda_min": 2.305666892618354e-22,
      "lambda_max": 0.06758096718337876,
      "normalized_lambda_min": 3.411710410065605e-21,
      "rank": 4
    },
    {
      "T": 0.5,
      "lambda_min": 1.1804620864251296e-19,
      "lambda_max": 0.1351918247242668,
      "normalized_lambda_min": 8.73175644187631e-19,
      "rank": 4
    },
    {
      "T": 1,
      "lambda_min": 6.043155805671134e-17,
      "lambda_max": 0.27062318792113427,
      "normalized_lambda_min": 2.233051739613771e-16,
      "rank": 6
    },
    {
      "T": 2,
      "lambda_min": 3.0924045672117935e-14,
      "lambda_max": 0.5431760901554346,
      "normalized_lambda_min": 5.693189783679313e-14,
      "rank": 6
    },
    {
      "T": 4,
      "lambda_min": 1.5795851897044102e-11,
      "lambda_max": 1.10223463146478,
      "normalized_lambda_min": 1.4330752678359146e-11,
      "rank": 8
    },
    {
      "T": 8,
      "lambda_min": 7.990126016306524e-09,
      "lambda_max": 2.3489309975259074,
      "normalized_lambda_min": 3.4016009941213256e-09,
      "rank": 9
    }
  ],
  "horizon_controllability_sweep": [
    {
      "design": "matched",
      "horizon": 1,
      "rank": 2,
      "min_nonzero_singular": 0.3044768777114372
    },
    {
      "design": "matched",
      "horizon": 2,
      "rank": 4,
      "min_nonzero_singular": 0.0357013246157235
    },
    {
      "design": "matched",
      "horizon": 3,
      "rank": 6,
      "min_nonzero_singular": 0.004088683386890841
    },
    {
      "design": "matched",
      "horizon": 4,
      "rank": 8,
      "min_nonzero_singular": 0.0006347821119892803
    },
    {
      "design": "matched",
      "horizon": 5,
      "rank": 10,
      "min_nonzero_singular": 3.920146419311282e-05
    },
    {
      "design": "matched",
      "horizon": 6,
      "rank": 10,
      "min_nonzero_singular": 3.920146419311282e-05
    },
    {
      "design": "matched",
      "horizon": 8,
      "rank": 10,
      "min_nonzero_singular": 3.920146419311282e-05
    },
    {
      "design": "matched",
      "horizon": 10,
      "rank": 10,
      "min_nonzero_singular": 3.920146419311282e-05
    },
    {
      "design": "unmatched",
      "horizon": 1,
      "rank": 1,
      "min_nonzero_singular": 0.3044768777114372
    },
    {
      "design": "unmatched",
      "horizon": 2,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "unmatched",
      "horizon": 3,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "unmatched",
      "horizon": 4,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "unmatched",
      "horizon": 5,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "unmatched",
      "horizon": 6,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "unmatched",
      "horizon": 8,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "unmatched",
      "horizon": 10,
      "rank": 2,
      "min_nonzero_singular": 0.2861480895156093
    },
    {
      "design": "inaccessible",
      "horizon": 1,
      "rank": 1,
      "min_nonzero_singular": 0.3044768777114372
    },
    {
      "design": "inaccessible",
      "horizon": 2,
      "rank": 2,
      "min_nonzero_singular": 0.04763093747028885
    },
    {
      "design": "inaccessible",
      "horizon": 3,
      "rank": 3,
      "min_nonzero_singular": 0.003576054447965049
    },
    {
      "design": "inaccessible",
      "horizon": 4,
      "rank": 4,
      "min_nonzero_singular": 0.001117511929278903
    },
    {
      "design": "inaccessible",
      "horizon": 5,
      "rank": 5,
      "min_nonzero_singular": 2.7173470102377014e-05
    },
    {
      "design": "inaccessible",
      "horizon": 6,
      "rank": 6,
      "min_nonzero_singular": 1.6382671428581099e-06
    },
    {
      "design": "inaccessible",
      "horizon": 8,
      "rank": 8,
      "min_nonzero_singular": 1.0726632771101887e-08
    },
    {
      "design": "inaccessible",
      "horizon": 10,
      "rank": 9,
      "min_nonzero_singular": 1.519551118083831e-09
    }
  ],
  "baseline_comparison": [
    {
      "design": "matching",
      "rank": 10,
      "lambda_min": 1.8061540249835195e-07
    },
    {
      "design": "random_sparse",
      "rank": 9,
      "lambda_min": 4.3144444864294774e-05
    },
    {
      "design": "matching",
      "rank": 7,
      "lambda_min": 2.8519836274807497e-13
    },
    {
      "design": "random_sparse",
      "rank": 10,
      "lambda_min": 3.197166506570611e-05
    },
    {
      "design": "matching",
      "rank": 10,
      "lambda_min": 8.347982762863937e-06
    },
    {
      "design": "random_sparse",
      "rank": 9,
      "lambda_min": 5.9121147045837824e-08
    },
    {
      "design": "matching",
      "rank": 10,
      "lambda_min": 3.1810950647869205e-06
    },
    {
      "design": "random_sparse",
      "rank": 10,
      "lambda_min": 0.0010977343123756381
    },
    {
      "design": "matching",
      "rank": 10,
      "lambda_min": 1.1218557849965952e-08
    },
    {
      "design": "random_sparse",
      "rank": 7,
      "lambda_min": 3.5961939339453158e-06
    },
    {
      "design": "matching",
      "rank": 9,
      "lambda_min": 1.3076005964932231e-09
    },
    {
      "design": "random_sparse",
      "rank": 9,
      "lambda_min": 1.1612567852377557e-05
    },
    {
      "design": "matching",
      "rank": 9,
      "lambda_min": 6.515699054929825e-08
    },
    {
      "design": "random_sparse",
      "rank": 9,
      "lambda_min": 0.00020769065247788058
    },
    {
      "design": "matching",
      "rank": 9,
      "lambda_min": 7.990126016306524e-09
    },
    {
      "design": "random_sparse",
      "rank": 8,
      "lambda_min": 3.3196816978292985e-05
    }
  ],
  "predictions": [
    "accessibility plus row-saturating matching predicts generic full controllability rank",
    "the matched finite-horizon Gramian has positive minimum eigenvalue for generic weights",
    "increasing horizon increases the minimum Gramian eigenvalue for the matched chain, while inaccessible remains singular"
  ]
}