Recorded data and provenance for the associated calculation.

Download recorded JSON

experiment_keyminimal-carrier-ground-proper-current-response
created_utc2026-10-12T09:21:46.744376+00:00
View complete data
{
  "experiment_key": "minimal-carrier-ground-proper-current-response",
  "created_utc": "2026-10-12T09:21:46.744376+00:00",
  "sources": {
    "experiments/minimal-carrier-ground-response.py": "be27aa03a9b7957c3ec8e6201bbff9977653d9a649e22d085ffc89c0d7c39bfd",
    "experiments/minimal-count-carrier.py": "c32f6a998a844bbccd3bc46483a551addf224b2b254f7b6d218a8fd0ca87ad2b",
    "experiments/forward-count-action.py": "5f1dd9a03cf8e6b07a56d507791fed8ff859f10abf22bb755c60c3bd1ed6d909",
    "experiments/inverse-current-radial.py": "52d9f1c981635192e06d96b11f0b1e4e5e2d1bf3d5ed420b59411fd45ef7be80",
    "experiments/excited-shell-orientation-current.py": "508484d445dbb86da9f88a31642268b5a1dba8264588e6f48ebb6cc00cf9c264"
  },
  "cases": [
    {
      "depth": 2,
      "minimal_columns": 2,
      "candidate_real_rank": 28,
      "training_closure_residual": 1.408361527271525e-15,
      "independent_closure_residual": 1.512676267336471e-15,
      "numerical_invariant_space_verified": true,
      "time_generator_eigenvalues": [
        {
          "real": -3.608224830031759e-16,
          "imaginary": 3.3333333333333366
        },
        {
          "real": -3.608224830031759e-16,
          "imaginary": -3.3333333333333366
        },
        {
          "real": 5.551115123125783e-17,
          "imaginary": 3.8542533774663803
        },
        {
          "real": 5.551115123125783e-17,
          "imaginary": -3.8542533774663803
        },
        {
          "real": 5.551115123125783e-16,
          "imaginary": 3.854253377466387
        },
        {
          "real": 5.551115123125783e-16,
          "imaginary": -3.854253377466387
        },
        {
          "real": 0.0,
          "imaginary": 3.8542533774663865
        },
        {
          "real": 0.0,
          "imaginary": -3.8542533774663865
        },
        {
          "real": 0.13116284589789418,
          "imaginary": 1.3527470934524586
        },
        {
          "real": 0.13116284589789418,
          "imaginary": -1.3527470934524586
        },
        {
          "real": 0.13116284589789423,
          "imaginary": 1.3527470934524617
        },
        {
          "real": 0.13116284589789423,
          "imaginary": -1.3527470934524617
        },
        {
          "real": 0.1311628458978939,
          "imaginary": 1.3527470934524646
        },
        {
          "real": 0.1311628458978939,
          "imaginary": -1.3527470934524646
        },
        {
          "real": -0.13116284589789212,
          "imaginary": 1.3527470934524628
        },
        {
          "real": -0.13116284589789212,
          "imaginary": -1.3527470934524628
        },
        {
          "real": -0.13116284589789512,
          "imaginary": 1.3527470934524621
        },
        {
          "real": -0.13116284589789512,
          "imaginary": -1.3527470934524621
        },
        {
          "real": -0.1311628458978953,
          "imaginary": 1.3527470934524626
        },
        {
          "real": -0.1311628458978953,
          "imaginary": -1.3527470934524626
        },
        {
          "real": 2.7006821409563363e-08,
          "imaginary": 0.0
        },
        {
          "real": -2.7006821093678996e-08,
          "imaginary": 0.0
        },
        {
          "real": -3.796533987330834e-16,
          "imaginary": 1.9906911580680044e-08
        },
        {
          "real": -3.796533987330834e-16,
          "imaginary": -1.9906911580680044e-08
        },
        {
          "real": -1.562414288851586e-08,
          "imaginary": 0.0
        },
        {
          "real": 1.5624142392752525e-08,
          "imaginary": 0.0
        },
        {
          "real": 4.390222657310068e-16,
          "imaginary": 1.4399577871509455e-16
        },
        {
          "real": 4.390222657310068e-16,
          "imaginary": -1.4399577871509455e-16
        }
      ],
      "maximum_growth_rate": 0.13116284589789423,
      "classification": "numerically verified finite ground spatial family"
    },
    {
      "depth": 3,
      "minimal_columns": 2,
      "candidate_real_rank": 36,
      "training_closure_residual": 1.5172910860365043e-15,
      "independent_closure_residual": 2.191621813497261e-15,
      "numerical_invariant_space_verified": true,
      "time_generator_eigenvalues": [
        {
          "real": 5.551115123125783e-16,
          "imaginary": 5.000000000000006
        },
        {
          "real": 5.551115123125783e-16,
          "imaginary": -5.000000000000006
        },
        {
          "real": -1.0477729794899915e-15,
          "imaginary": 5.620094311597661
        },
        {
          "real": -1.0477729794899915e-15,
          "imaginary": -5.620094311597661
        },
        {
          "real": 1.3322676295501878e-15,
          "imaginary": 5.620094311597657
        },
        {
          "real": 1.3322676295501878e-15,
          "imaginary": -5.620094311597657
        },
        {
          "real": -2.220446049250313e-16,
          "imaginary": 5.620094311597655
        },
        {
          "real": -2.220446049250313e-16,
          "imaginary": -5.620094311597655
        },
        {
          "real": -2.220446049250313e-16,
          "imaginary": 5.6200943115976445
        },
        {
          "real": -2.220446049250313e-16,
          "imaginary": -5.6200943115976445
        },
        {
          "real": -8.326672684688674e-16,
          "imaginary": 2.108636169925726
        },
        {
          "real": -8.326672684688674e-16,
          "imaginary": -2.108636169925726
        },
        {
          "real": 3.771288836773579e-15,
          "imaginary": 2.108636169925719
        },
        {
          "real": 3.771288836773579e-15,
          "imaginary": -2.108636169925719
        },
        {
          "real": 6.661338147750939e-16,
          "imaginary": 2.1086361699257212
        },
        {
          "real": 6.661338147750939e-16,
          "imaginary": -2.1086361699257212
        },
        {
          "real": 1.3877787807814457e-16,
          "imaginary": 2.108636169925718
        },
        {
          "real": 1.3877787807814457e-16,
          "imaginary": -2.108636169925718
        },
        {
          "real": 6.106226635438361e-16,
          "imaginary": 1.6876591574214928
        },
        {
          "real": 6.106226635438361e-16,
          "imaginary": -1.6876591574214928
        },
        {
          "real": -1.790234627208065e-15,
          "imaginary": 1.6876591574215023
        },
        {
          "real": -1.790234627208065e-15,
          "imaginary": -1.6876591574215023
        },
        {
          "real": 0.0,
          "imaginary": 1.6876591574214976
        },
        {
          "real": 0.0,
          "imaginary": -1.6876591574214976
        },
        {
          "real": 1.2212453270876722e-15,
          "imaginary": 1.6876591574215
        },
        {
          "real": 1.2212453270876722e-15,
          "imaginary": -1.6876591574215
        },
        {
          "real": -1.2665347938019486e-16,
          "imaginary": 9.461306238082524e-16
        },
        {
          "real": -1.2665347938019486e-16,
          "imaginary": -9.461306238082524e-16
        },
        {
          "real": 9.143908108398276e-17,
          "imaginary": 6.216460805302504e-16
        },
        {
          "real": 9.143908108398276e-17,
          "imaginary": -6.216460805302504e-16
        },
        {
          "real": -6.980039657090549e-16,
          "imaginary": 0.0
        },
        {
          "real": 3.727082349715275e-16,
          "imaginary": 0.0
        },
        {
          "real": 9.854606128306926e-17,
          "imaginary": 0.0
        },
        {
          "real": -2.1706193222706e-16,
          "imaginary": 2.1530029373485181e-16
        },
        {
          "real": -2.1706193222706e-16,
          "imaginary": -2.1530029373485181e-16
        },
        {
          "real": -2.082219802915955e-16,
          "imaginary": 0.0
        }
      ],
      "maximum_growth_rate": 3.771288836773579e-15,
      "classification": "numerically verified finite ground spatial family"
    },
    {
      "depth": 4,
      "minimal_columns": 4,
      "candidate_real_rank": 44,
      "training_closure_residual": 1.427440940732062e-15,
      "independent_closure_residual": 2.7813671921338157e-15,
      "numerical_invariant_space_verified": true,
      "time_generator_eigenvalues": [
        {
          "real": 0.0,
          "imaginary": 6.66666666666666
        },
        {
          "real": 0.0,
          "imaginary": -6.66666666666666
        },
        {
          "real": -1.3322676295501878e-15,
          "imaginary": 7.278189360771191
        },
        {
          "real": -1.3322676295501878e-15,
          "imaginary": -7.278189360771191
        },
        {
          "real": -4.6629367034256575e-15,
          "imaginary": 7.2781893607711945
        },
        {
          "real": -4.6629367034256575e-15,
          "imaginary": -7.2781893607711945
        },
        {
          "real": 1.7763568394002505e-15,
          "imaginary": 7.278189360771198
        },
        {
          "real": 1.7763568394002505e-15,
          "imaginary": -7.278189360771198
        },
        {
          "real": 2.886579864025407e-15,
          "imaginary": 7.278189360771204
        },
        {
          "real": 2.886579864025407e-15,
          "imaginary": -7.278189360771204
        },
        {
          "real": -1.9984014443252818e-15,
          "imaginary": 7.278189360771207
        },
        {
          "real": -1.9984014443252818e-15,
          "imaginary": -7.278189360771207
        },
        {
          "real": 0.21964929620853463,
          "imaginary": 2.128766743897876
        },
        {
          "real": 0.21964929620853463,
          "imaginary": -2.128766743897876
        },
        {
          "real": 0.2196492962085354,
          "imaginary": 2.128766743897892
        },
        {
          "real": 0.2196492962085354,
          "imaginary": -2.128766743897892
        },
        {
          "real": 0.21964929620854262,
          "imaginary": 2.1287667438978795
        },
        {
          "real": 0.21964929620854262,
          "imaginary": -2.1287667438978795
        },
        {
          "real": 0.219649296208548,
          "imaginary": 2.128766743897884
        },
        {
          "real": 0.219649296208548,
          "imaginary": -2.128766743897884
        },
        {
          "real": 0.21964929620854765,
          "imaginary": 2.128766743897891
        },
        {
          "real": 0.21964929620854765,
          "imaginary": -2.128766743897891
        },
        {
          "real": -0.21964929620853313,
          "imaginary": 2.1287667438978977
        },
        {
          "real": -0.21964929620853313,
          "imaginary": -2.1287667438978977
        },
        {
          "real": -0.21964929620853463,
          "imaginary": 2.128766743897882
        },
        {
          "real": -0.21964929620853463,
          "imaginary": -2.128766743897882
        },
        {
          "real": -0.2196492962085483,
          "imaginary": 2.1287667438978874
        },
        {
          "real": -0.2196492962085483,
          "imaginary": -2.1287667438978874
        },
        {
          "real": -0.2196492962085445,
          "imaginary": 2.1287667438978897
        },
        {
          "real": -0.2196492962085445,
          "imaginary": -2.1287667438978897
        },
        {
          "real": -0.21964929620854695,
          "imaginary": 2.1287667438978826
        },
        {
          "real": -0.21964929620854695,
          "imaginary": -2.1287667438978826
        },
        {
          "real": -1.1511976822011232e-16,
          "imaginary": 1.0797468852673738e-15
        },
        {
          "real": -1.1511976822011232e-16,
          "imaginary": -1.0797468852673738e-15
        },
        {
          "real": 9.471244893150604e-16,
          "imaginary": 0.0
        },
        {
          "real": -4.2416733786246574e-16,
          "imaginary": 7.000349024500293e-16
        },
        {
          "real": -4.2416733786246574e-16,
          "imaginary": -7.000349024500293e-16
        },
        {
          "real": 2.5015526497949264e-16,
          "imaginary": 3.8100906765013835e-16
        },
        {
          "real": 2.5015526497949264e-16,
          "imaginary": -3.8100906765013835e-16
        },
        {
          "real": 3.244410444006069e-16,
          "imaginary": 0.0
        },
        {
          "real": -1.9615641058045948e-16,
          "imaginary": 2.1898621667519923e-16
        },
        {
          "real": -1.9615641058045948e-16,
          "imaginary": -2.1898621667519923e-16
        },
        {
          "real": -4.656254865866845e-16,
          "imaginary": 2.251497410700889e-16
        },
        {
          "real": -4.656254865866845e-16,
          "imaginary": -2.251497410700889e-16
        }
      ],
      "maximum_growth_rate": 0.219649296208548,
      "classification": "numerically verified finite ground spatial family"
    },
    {
      "depth": 5,
      "minimal_columns": 1,
      "candidate_real_rank": 80,
      "training_closure_residual": 1.7332775228738119e-15,
      "independent_closure_residual": 2.9976423546834852e-15,
      "numerical_invariant_space_verified": true,
      "time_generator_eigenvalues": [
        {
          "real": 8.881784197001252e-16,
          "imaginary": 8.333333333333321
        },
        {
          "real": 8.881784197001252e-16,
          "imaginary": -8.333333333333321
        },
        {
          "real": -3.9968028886505635e-15,
          "imaginary": 8.85948042334753
        },
        {
          "real": -3.9968028886505635e-15,
          "imaginary": -8.85948042334753
        },
        {
          "real": 3.774758283725532e-15,
          "imaginary": 8.859480423347595
        },
        {
          "real": 3.774758283725532e-15,
          "imaginary": -8.859480423347595
        },
        {
          "real": 0.0,
          "imaginary": 8.859480423347566
        },
        {
          "real": 0.0,
          "imaginary": -8.859480423347566
        },
        {
          "real": 4.440892098500626e-16,
          "imaginary": 8.859480423347545
        },
        {
          "real": 4.440892098500626e-16,
          "imaginary": -8.859480423347545
        },
        {
          "real": -1.214306433183765e-15,
          "imaginary": 5.272847400411784
        },
        {
          "real": -1.214306433183765e-15,
          "imaginary": -5.272847400411784
        },
        {
          "real": 7.771561172376096e-16,
          "imaginary": 8.859480423347573
        },
        {
          "real": 7.771561172376096e-16,
          "imaginary": -8.859480423347573
        },
        {
          "real": -5.551115123125783e-16,
          "imaginary": 8.859480423347561
        },
        {
          "real": -5.551115123125783e-16,
          "imaginary": -8.859480423347561
        },
        {
          "real": 8.326672684688674e-15,
          "imaginary": 5.809721347698848
        },
        {
          "real": 8.326672684688674e-15,
          "imaginary": -5.809721347698848
        },
        {
          "real": 1.7763568394002505e-15,
          "imaginary": 5.809721347698874
        },
        {
          "real": 1.7763568394002505e-15,
          "imaginary": -5.809721347698874
        },
        {
          "real": -3.1086244689504383e-15,
          "imaginary": 5.809721347698855
        },
        {
          "real": -3.1086244689504383e-15,
          "imaginary": -5.809721347698855
        },
        {
          "real": 3.552713678800501e-15,
          "imaginary": 5.809721347698875
        },
        {
          "real": 3.552713678800501e-15,
          "imaginary": -5.809721347698875
        },
        {
          "real": 1.3322676295501878e-15,
          "imaginary": 5.8097213476988605
        },
        {
          "real": 1.3322676295501878e-15,
          "imaginary": -5.8097213476988605
        },
        {
          "real": 1.5543122344752192e-15,
          "imaginary": 5.8097213476988605
        },
        {
          "real": 1.5543122344752192e-15,
          "imaginary": -5.8097213476988605
        },
        {
          "real": -0.623035390290446,
          "imaginary": 1.7780244226415256
        },
        {
          "real": -0.623035390290446,
          "imaginary": -1.7780244226415256
        },
        {
          "real": -0.6230353902904571,
          "imaginary": 1.778024422641525
        },
        {
          "real": -0.6230353902904571,
          "imaginary": -1.778024422641525
        },
        {
          "real": -0.6230353902904541,
          "imaginary": 1.7780244226415285
        },
        {
          "real": -0.6230353902904541,
          "imaginary": -1.7780244226415285
        },
        {
          "real": -0.6230353902904512,
          "imaginary": 1.77802442264153
        },
        {
          "real": -0.6230353902904512,
          "imaginary": -1.77802442264153
        },
        {
          "real": -0.6230353902904567,
          "imaginary": 1.7780244226415307
        },
        {
          "real": -0.6230353902904567,
          "imaginary": -1.7780244226415307
        },
        {
          "real": 0.6230353902904568,
          "imaginary": 1.7780244226415232
        },
        {
          "real": 0.6230353902904568,
          "imaginary": -1.7780244226415232
        },
        {
          "real": 0.6230353902904525,
          "imaginary": 1.778024422641523
        },
        {
          "real": 0.6230353902904525,
          "imaginary": -1.778024422641523
        },
        {
          "real": 0.6230353902904504,
          "imaginary": 1.778024422641532
        },
        {
          "real": 0.6230353902904504,
          "imaginary": -1.778024422641532
        },
        {
          "real": 1.9290125052862095e-15,
          "imaginary": 2.282788547754721
        },
        {
          "real": 1.9290125052862095e-15,
          "imaginary": -2.282788547754721
        },
        {
          "real": 2.1094237467877974e-15,
          "imaginary": 2.282788547754725
        },
        {
          "real": 2.1094237467877974e-15,
          "imaginary": -2.282788547754725
        },
        {
          "real": -1.27675647831893e-15,
          "imaginary": 2.2827885477547225
        },
        {
          "real": -1.27675647831893e-15,
          "imaginary": -2.2827885477547225
        },
        {
          "real": 0.6230353902904553,
          "imaginary": 1.7780244226415283
        },
        {
          "real": 0.6230353902904553,
          "imaginary": -1.7780244226415283
        },
        {
          "real": 9.71445146547012e-16,
          "imaginary": 2.282788547754725
        },
        {
          "real": 9.71445146547012e-16,
          "imaginary": -2.282788547754725
        },
        {
          "real": 0.623035390290454,
          "imaginary": 1.7780244226415203
        },
        {
          "real": 0.623035390290454,
          "imaginary": -1.7780244226415203
        },
        {
          "real": 3.0253577421035516e-15,
          "imaginary": 2.2827885477547327
        },
        {
          "real": 3.0253577421035516e-15,
          "imaginary": -2.2827885477547327
        },
        {
          "real": 0.6230353902904536,
          "imaginary": 1.7780244226415332
        },
        {
          "real": 0.6230353902904536,
          "imaginary": -1.7780244226415332
        },
        {
          "real": -3.774758283725532e-15,
          "imaginary": 2.282788547754733
        },
        {
          "real": -3.774758283725532e-15,
          "imaginary": -2.282788547754733
        },
        {
          "real": -0.623035390290452,
          "imaginary": 1.7780244226415294
        },
        {
          "real": -0.623035390290452,
          "imaginary": -1.7780244226415294
        },
        {
          "real": -8.309139385041906e-08,
          "imaginary": 1.505717327949794e-08
        },
        {
          "real": -8.309139385041906e-08,
          "imaginary": -1.505717327949794e-08
        },
        {
          "real": -7.856454673211207e-08,
          "imaginary": 0.0
        },
        {
          "real": -5.179024175491891e-08,
          "imaginary": 4.8725543600368594e-08
        },
        {
          "real": -5.179024175491891e-08,
          "imaginary": -4.8725543600368594e-08
        },
        {
          "real": 2.371261370211546e-15,
          "imaginary": 6.077799242401157e-08
        },
        {
          "real": 2.371261370211546e-15,
          "imaginary": -6.077799242401157e-08
        },
        {
          "real": 5.1790239246043334e-08,
          "imaginary": 4.8725542470855466e-08
        },
        {
          "real": 5.1790239246043334e-08,
          "imaginary": -4.8725542470855466e-08
        },
        {
          "real": 1.6276855163403163e-15,
          "imaginary": 4.132841599356728e-08
        },
        {
          "real": 1.6276855163403163e-15,
          "imaginary": -4.132841599356728e-08
        },
        {
          "real": 8.30913920027468e-08,
          "imaginary": 1.5057178106245253e-08
        },
        {
          "real": 8.30913920027468e-08,
          "imaginary": -1.5057178106245253e-08
        },
        {
          "real": 7.856454809773298e-08,
          "imaginary": 0.0
        },
        {
          "real": -4.8920449182241475e-16,
          "imaginary": 0.0
        },
        {
          "real": 1.3551320582269866e-15,
          "imaginary": 0.0
        }
      ],
      "maximum_growth_rate": 0.6230353902904568,
      "classification": "numerically verified finite ground spatial family"
    },
    {
      "depth": 6,
      "minimal_columns": 1,
      "candidate_real_rank": 60,
      "training_closure_residual": 1.5934946658728355e-15,
      "independent_closure_residual": 2.965872402153402e-15,
      "numerical_invariant_space_verified": true,
      "time_generator_eigenvalues": [
        {
          "real": 9.769962616701378e-15,
          "imaginary": 9.999999999999998
        },
        {
          "real": 9.769962616701378e-15,
          "imaginary": -9.999999999999998
        },
        {
          "real": -3.9968028886505635e-15,
          "imaginary": 10.490944691595445
        },
        {
          "real": -3.9968028886505635e-15,
          "imaginary": -10.490944691595445
        },
        {
          "real": -3.552713678800501e-15,
          "imaginary": 10.490944691595448
        },
        {
          "real": -3.552713678800501e-15,
          "imaginary": -10.490944691595448
        },
        {
          "real": -3.552713678800501e-15,
          "imaginary": 10.490944691595457
        },
        {
          "real": -3.552713678800501e-15,
          "imaginary": -10.490944691595457
        },
        {
          "real": 2.6645352591003757e-15,
          "imaginary": 10.490944691595454
        },
        {
          "real": 2.6645352591003757e-15,
          "imaginary": -10.490944691595454
        },
        {
          "real": 0.0,
          "imaginary": 10.490944691595477
        },
        {
          "real": 0.0,
          "imaginary": -10.490944691595477
        },
        {
          "real": 1.7763568394002505e-15,
          "imaginary": 10.490944691595484
        },
        {
          "real": 1.7763568394002505e-15,
          "imaginary": -10.490944691595484
        },
        {
          "real": -1.3322676295501878e-15,
          "imaginary": 10.490944691595491
        },
        {
          "real": -1.3322676295501878e-15,
          "imaginary": -10.490944691595491
        },
        {
          "real": -0.7366352313768668,
          "imaginary": 1.863081283636636
        },
        {
          "real": -0.7366352313768668,
          "imaginary": -1.863081283636636
        },
        {
          "real": -0.73663523137687,
          "imaginary": 1.8630812836366328
        },
        {
          "real": -0.73663523137687,
          "imaginary": -1.8630812836366328
        },
        {
          "real": -0.7366352313768763,
          "imaginary": 1.8630812836366397
        },
        {
          "real": -0.7366352313768763,
          "imaginary": -1.8630812836366397
        },
        {
          "real": -0.7366352313768761,
          "imaginary": 1.863081283636639
        },
        {
          "real": -0.7366352313768761,
          "imaginary": -1.863081283636639
        },
        {
          "real": -0.7366352313768728,
          "imaginary": 1.8630812836366386
        },
        {
          "real": -0.7366352313768728,
          "imaginary": -1.8630812836366386
        },
        {
          "real": -0.7366352313768738,
          "imaginary": 1.863081283636638
        },
        {
          "real": -0.7366352313768738,
          "imaginary": -1.863081283636638
        },
        {
          "real": -0.7366352313768737,
          "imaginary": 1.8630812836366322
        },
        {
          "real": -0.7366352313768737,
          "imaginary": -1.8630812836366322
        },
        {
          "real": 0.7366352313768731,
          "imaginary": 1.863081283636643
        },
        {
          "real": 0.7366352313768731,
          "imaginary": -1.863081283636643
        },
        {
          "real": 0.7366352313768714,
          "imaginary": 1.8630812836366344
        },
        {
          "real": 0.7366352313768714,
          "imaginary": -1.8630812836366344
        },
        {
          "real": 0.7366352313768749,
          "imaginary": 1.8630812836366373
        },
        {
          "real": 0.7366352313768749,
          "imaginary": -1.8630812836366373
        },
        {
          "real": 0.7366352313768731,
          "imaginary": 1.8630812836366417
        },
        {
          "real": 0.7366352313768731,
          "imaginary": -1.8630812836366417
        },
        {
          "real": 0.7366352313768711,
          "imaginary": 1.863081283636642
        },
        {
          "real": 0.7366352313768711,
          "imaginary": -1.863081283636642
        },
        {
          "real": 0.7366352313768714,
          "imaginary": 1.8630812836366342
        },
        {
          "real": 0.7366352313768714,
          "imaginary": -1.8630812836366342
        },
        {
          "real": 0.7366352313768687,
          "imaginary": 1.8630812836366342
        },
        {
          "real": 0.7366352313768687,
          "imaginary": -1.8630812836366342
        },
        {
          "real": 7.571467522371921e-08,
          "imaginary": 1.1206359341781642e-08
        },
        {
          "real": 7.571467522371921e-08,
          "imaginary": -1.1206359341781642e-08
        },
        {
          "real": 2.5264543578558766e-08,
          "imaginary": 1.1543218772096443e-07
        },
        {
          "real": 2.5264543578558766e-08,
          "imaginary": -1.1543218772096443e-07
        },
        {
          "real": 3.605388747812399e-15,
          "imaginary": 1.0610956836139245e-07
        },
        {
          "real": 3.605388747812399e-15,
          "imaginary": -1.0610956836139245e-07
        },
        {
          "real": -2.526454955525363e-08,
          "imaginary": 1.1543218928368525e-07
        },
        {
          "real": -2.526454955525363e-08,
          "imaginary": -1.1543218928368525e-07
        },
        {
          "real": 3.177594903724241e-15,
          "imaginary": 7.329837896698286e-08
        },
        {
          "real": 3.177594903724241e-15,
          "imaginary": -7.329837896698286e-08
        },
        {
          "real": -7.571467421235902e-08,
          "imaginary": 1.1206358051016415e-08
        },
        {
          "real": -7.571467421235902e-08,
          "imaginary": -1.1206358051016415e-08
        },
        {
          "real": 8.46185646486977e-09,
          "imaginary": 0.0
        },
        {
          "real": -8.461787708251352e-09,
          "imaginary": 0.0
        },
        {
          "real": 6.2307640787070415e-15,
          "imaginary": 0.0
        },
        {
          "real": -7.621923558365666e-14,
          "imaginary": 0.0
        }
      ],
      "maximum_growth_rate": 0.7366352313768749,
      "classification": "numerically verified finite ground spatial family"
    },
    {
      "depth": 10,
      "minimal_columns": 1,
      "candidate_real_rank": 92,
      "training_closure_residual": 0.20408748806525645,
      "independent_closure_residual": 0.15217560962398602,
      "numerical_invariant_space_verified": false,
      "time_generator_eigenvalues": [],
      "maximum_growth_rate": null,
      "classification": "candidate family leaks; projected eigenvalues are not reported"
    }
  ],
  "limits": [
    "the operator uses the affine proper-current chart with c3/5 and epsilon=c/Hd",
    "only independent numerical closure is claimed at this stage; exact coefficient proof remains separate",
    "these are ground controls, not assigned excited particle states",
    "a closed-family growing eigenvalue would disprove linear persistence of that preparation",
    "absence of growth in this family would not establish complete angular or nonlinear persistence"
  ]
}