{
  "seed": 1468,
  "lambda": 1e-08,
  "prediction_1_identity_rank": {
    "predicted": "ico rank=6; coordinate rank=3",
    "observed": {
      "ico_rank": 6,
      "coordinate_rank": 3,
      "ico_sigma_min": 0.8944271909999157,
      "coordinate_smallest_singular_value": 0.0
    }
  },
  "prediction_2_rank_preservation": {
    "predicted": "ico rank remains 6 for every invertible F; coordinate remains rank 3",
    "observed": [
      {
        "anisotropy": 0.0,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.8944271909999157,
        "coordinate_smallest_nonzero_singular_value": 1.0,
        "coordinate_has_zero_singular_values": true
      },
      {
        "anisotropy": 0.5,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.5424975142220966,
        "coordinate_smallest_nonzero_singular_value": 0.7094545301398459,
        "coordinate_has_zero_singular_values": true
      },
      {
        "anisotropy": 1.0,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.32904137519359206,
        "coordinate_smallest_nonzero_singular_value": 0.481646139494879,
        "coordinate_has_zero_singular_values": true
      },
      {
        "anisotropy": 1.5,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.19957368236892148,
        "coordinate_smallest_nonzero_singular_value": 0.22372305543972865,
        "coordinate_has_zero_singular_values": true
      },
      {
        "anisotropy": 2.0,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.12104755722850132,
        "coordinate_smallest_nonzero_singular_value": 0.5757695402039034,
        "coordinate_has_zero_singular_values": true
      },
      {
        "anisotropy": 2.5,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.07341905474240482,
        "coordinate_smallest_nonzero_singular_value": 0.946437150034156,
        "coordinate_has_zero_singular_values": true
      },
      {
        "anisotropy": 3.0,
        "ico_rank": 6,
        "coordinate_rank": 3,
        "ico_sigma_min": 0.04453090770838472,
        "coordinate_smallest_nonzero_singular_value": 1.2625973447598366,
        "coordinate_has_zero_singular_values": true
      }
    ]
  },
  "prediction_3_noise_conditioning": {
    "predicted": "noise error increases with inverse smallest singular value; coordinate is non-identifiable",
    "noise_std": 0.001,
    "trials": 100,
    "observed": [
      {
        "anisotropy": 0.0,
        "icosahedral": {
          "relative_error_mean": 0.0012704696432291392,
          "relative_error_std": 0.000595841096511093,
          "inv_sigma_min_mean": 1.1180339887498953,
          "condition_mean": 1.58113883008419
        },
        "random6": {
          "relative_error_mean": 0.010035117205568666,
          "relative_error_std": 0.007045079649234555,
          "inv_sigma_min_mean": 29.106303243105593,
          "condition_mean": 55.40525057140209
        },
        "coordinate": {
          "relative_error_mean": 0.7509031535351902,
          "relative_error_std": 0.19237631049438733,
          "inv_sigma_min_mean": 1.0,
          "condition_mean": 1.0
        }
      },
      {
        "anisotropy": 0.5,
        "icosahedral": {
          "relative_error_mean": 0.0016759503168395102,
          "relative_error_std": 0.0009917374340478293,
          "inv_sigma_min_mean": 1.84332641861766,
          "condition_mean": 5.520952752916615
        },
        "random6": {
          "relative_error_mean": 0.011727149276517907,
          "relative_error_std": 0.008511823816994935,
          "inv_sigma_min_mean": 34.78187676411992,
          "condition_mean": 107.1529411243173
        },
        "coordinate": {
          "relative_error_mean": 0.7064669842638811,
          "relative_error_std": 0.1677456386351106,
          "inv_sigma_min_mean": 1.4131901704912775,
          "condition_mean": 3.0640291801418793
        }
      },
      {
        "anisotropy": 1.0,
        "icosahedral": {
          "relative_error_mean": 0.001955061256420879,
          "relative_error_std": 0.0010576025339922874,
          "inv_sigma_min_mean": 3.0391314752184244,
          "condition_mean": 24.606451260891898
        },
        "random6": {
          "relative_error_mean": 0.01953033383886621,
          "relative_error_std": 0.021525895369767734,
          "inv_sigma_min_mean": 55.14237410610069,
          "condition_mean": 451.59089847290375
        },
        "coordinate": {
          "relative_error_mean": 0.6601241757105272,
          "relative_error_std": 0.21227515513759707,
          "inv_sigma_min_mean": 1.7746943455139106,
          "condition_mean": 10.513500543961538
        }
      },
      {
        "anisotropy": 1.5,
        "icosahedral": {
          "relative_error_mean": 0.0035073731163582245,
          "relative_error_std": 0.0021749712428775505,
          "inv_sigma_min_mean": 5.01068070764688,
          "condition_mean": 110.24953183759459
        },
        "random6": {
          "relative_error_mean": 0.03586979694229784,
          "relative_error_std": 0.04894632423397871,
          "inv_sigma_min_mean": 91.08716803115443,
          "condition_mean": 2039.831902718666
        },
        "coordinate": {
          "relative_error_mean": 0.6409480505081112,
          "relative_error_std": 0.19058423030337004,
          "inv_sigma_min_mean": 1.9217242575316122,
          "condition_mean": 31.336629130650948
        }
      },
      {
        "anisotropy": 2.0,
        "icosahedral": {
          "relative_error_mean": 0.0060951171569506605,
          "relative_error_std": 0.004521106462034513,
          "inv_sigma_min_mean": 8.261215863384216,
          "condition_mean": 494.0976831195629
        },
        "random6": {
          "relative_error_mean": 0.06118769093632997,
          "relative_error_std": 0.07527994978442538,
          "inv_sigma_min_mean": 148.11066890806288,
          "condition_mean": 8845.367386430931
        },
        "coordinate": {
          "relative_error_mean": 0.6127052025895047,
          "relative_error_std": 0.1946151960050764,
          "inv_sigma_min_mean": 2.014373407204584,
          "condition_mean": 89.6416797921853
        }
      },
      {
        "anisotropy": 2.5,
        "icosahedral": {
          "relative_error_mean": 0.010416620657133549,
          "relative_error_std": 0.007280759986288966,
          "inv_sigma_min_mean": 13.6204423158071,
          "condition_mean": 2214.3907496289157
        },
        "random6": {
          "relative_error_mean": 0.0983556860757841,
          "relative_error_std": 0.08012111424667348,
          "inv_sigma_min_mean": 238.66808383508354,
          "condition_mean": 36154.15117052386
        },
        "coordinate": {
          "relative_error_mean": 0.6190785925765124,
          "relative_error_std": 0.1987391274144127,
          "inv_sigma_min_mean": 2.1607479661814075,
          "condition_mean": 259.1801471278394
        }
      },
      {
        "anisotropy": 3.0,
        "icosahedral": {
          "relative_error_mean": 0.016115070630707527,
          "relative_error_std": 0.010436245198162317,
          "inv_sigma_min_mean": 22.45631296241717,
          "condition_mean": 9924.210499549661
        },
        "random6": {
          "relative_error_mean": 0.1688605522348797,
          "relative_error_std": 0.13056222189022024,
          "inv_sigma_min_mean": 408.0125935163771,
          "condition_mean": 188053.61435777947
        },
        "coordinate": {
          "relative_error_mean": 0.6012848091403088,
          "relative_error_std": 0.20733236015966489,
          "inv_sigma_min_mean": 2.5483234784416027,
          "condition_mean": 836.6397187169674
        }
      }
    ]
  },
  "exact_reconstruction_max_relative_error": {
    "icosahedral": 1.781274422149397e-09,
    "random6": 8.47308806937353e-07
  },
  "quantitative_predictions": {
    "P1_identity_rank": {
      "predicted_icosa_rank": 6,
      "observed_icosa_rank": 6,
      "predicted_coordinate_rank": 3,
      "observed_coordinate_rank": 3
    },
    "P2_SL3_rank_invariance": {
      "predicted_icosa_rank_at_all_strengths": 6,
      "observed_icosa_ranks": [
        6,
        6,
        6,
        6,
        6,
        6,
        6
      ],
      "predicted_coordinate_rank_at_all_strengths": 3,
      "observed_coordinate_ranks": [
        3,
        3,
        3,
        3,
        3,
        3,
        3
      ]
    },
    "P3_noise_amplification": {
      "prediction": "error rises as inverse smallest singular value rises",
      "strengths": [
        0.0,
        0.5,
        1.0,
        1.5,
        2.0,
        2.5,
        3.0
      ],
      "ico_error_at_strength_0_and_3": [
        0.0012704696432291392,
        0.016115070630707527
      ],
      "ico_inverse_sigma_at_strength_0_and_3": [
        1.1180339887498953,
        22.45631296241717
      ],
      "error_growth_factor": 12.684341350925964,
      "inverse_sigma_growth_factor": 20.08553692318978
    }
  },
  "summary": {
    "identity_ico_full_rank": true,
    "identity_coordinate_deficient": true,
    "all_ico_swept_full_rank": true,
    "all_coordinate_swept_deficient": true,
    "exact_ico_error": 1.781274422149397e-09,
    "exact_random6_error": 8.47308806937353e-07
  }
}