{
  "scope": "Necessary positional certificates, not an opacity decision procedure and not a solution of the minimum opaque-barrier problem.",
  "arithmetic": "Exact formulas evaluated in float64; numerical tests do not replace the analytic proofs.",
  "runtime": {
    "python": "3.12.14",
    "numpy": "2.3.5",
    "scipy": "1.17.0"
  },
  "summary": {
    "all_assertions_passed": true,
    "analytic_cases": 6,
    "random_cases": 100,
    "positive_collinear_overlap_rejected": true,
    "max_random_quadrature_discrepancy": 2.6645352591003757e-15,
    "max_random_symmetry_discrepancy": 2.6645352591003757e-15,
    "actual_opaque_examples": 3
  },
  "analytic_cases": [
    {
      "case": "parallel unit segments, separation 1",
      "expected": 0.8284271247461903,
      "endpoint_formula": 0.8284271247461903,
      "independent_quadrature": 0.8284271247461898,
      "quadrature_reported_error": 9.197388681172368e-15
    },
    {
      "case": "crossing orthogonal diameters of length 2",
      "expected": 2.3431457505076194,
      "endpoint_formula": 2.3431457505076194,
      "independent_quadrature": 2.34314575050762,
      "quadrature_reported_error": 2.601414362266152e-14
    },
    {
      "case": "right-triangle adjacent legs",
      "expected": 0.5857864376269049,
      "endpoint_formula": 0.5857864376269049,
      "independent_quadrature": 0.5857864376269049,
      "quadrature_reported_error": 6.50353590566538e-15
    },
    {
      "case": "collinear shared endpoint",
      "expected": 0.0,
      "endpoint_formula": 0.0,
      "independent_quadrature": 0.0,
      "quadrature_reported_error": 0.0
    },
    {
      "case": "collinear disjoint segments",
      "expected": 0.0,
      "endpoint_formula": 0.0,
      "independent_quadrature": 0.0,
      "quadrature_reported_error": 0.0
    },
    {
      "case": "degenerate isolated point",
      "expected": 0.0,
      "endpoint_formula": 0.0,
      "independent_quadrature": 0.0,
      "quadrature_reported_error": 0.0
    }
  ],
  "random_cases": [
    {
      "index": 0,
      "first": [
        [
          -0.4468469859068067,
          2.7638224228105006
        ],
        [
          2.867691333756259,
          2.1711630614953714
        ]
      ],
      "second": [
        [
          1.8217831760226755,
          -0.03546857480273591
        ],
        [
          -1.865922256578637,
          1.6394356559868095
        ]
      ],
      "endpoint_formula": 4.114051712773268,
      "independent_quadrature": 4.114051712773268,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.567514936019443e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 1,
      "first": [
        [
          -0.9422919322282626,
          -0.39994278185699494
        ],
        [
          0.2635127758732745,
          -2.939837425261449
        ]
      ],
      "second": [
        [
          1.8213007391904954,
          0.1654399137484095
        ],
        [
          2.8950233868704753,
          0.5645690904161733
        ]
      ],
      "endpoint_formula": 0.24685075102487097,
      "independent_quadrature": 0.2468507510248709,
      "absolute_discrepancy": 8.326672684688674e-17,
      "quadrature_reported_error": 2.7405938743382363e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 2,
      "first": [
        [
          -2.5302051017883196,
          0.6340877601802677
        ],
        [
          -1.3821077816646457,
          -2.052470391203479
        ]
      ],
      "second": [
        [
          -0.15329506148451788,
          0.373501119184819
        ],
        [
          1.4326472753283053,
          -0.05699406593372469
        ]
      ],
      "endpoint_formula": 0.9006124413018495,
      "independent_quadrature": 0.9006124413018495,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 9.998806685971857e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 3,
      "first": [
        [
          1.6108631063212284,
          -0.3773317904741962
        ],
        [
          -2.242532703755285,
          -2.9474046206822746
        ]
      ],
      "second": [
        [
          2.2531276208907434,
          -1.9114750527953805
        ],
        [
          2.640498777423268,
          -0.5111477630880463
        ]
      ],
      "endpoint_formula": 1.4684391607510996,
      "independent_quadrature": 1.4684391607510991,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 1.6302949665271118e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 4,
      "first": [
        [
          0.21289816673486772,
          -1.8076030653570316
        ],
        [
          -2.1592409470422966,
          -2.2164074648705423
        ]
      ],
      "second": [
        [
          -0.07954183619551047,
          2.397826403531048
        ],
        [
          -1.0530730736415943,
          -1.6666077181040249
        ]
      ],
      "endpoint_formula": 1.5910272670602037,
      "independent_quadrature": 1.5910272670602037,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 1.766395104696676e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 5,
      "first": [
        [
          -2.096278766890631,
          1.6825148117072946
        ],
        [
          2.5393374916133826,
          -0.13502094019032418
        ]
      ],
      "second": [
        [
          -2.855941808134772,
          -0.34514583863946147
        ],
        [
          2.1297847713373637,
          0.4709607481214939
        ]
      ],
      "endpoint_formula": 7.370179349753792,
      "independent_quadrature": 7.37017934975379,
      "absolute_discrepancy": 2.6645352591003757e-15,
      "quadrature_reported_error": 8.182542809713521e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 6,
      "first": [
        [
          -2.67258470176568,
          0.6546835040874086
        ],
        [
          -1.3980563958592331,
          -1.4655561374187347
        ]
      ],
      "second": [
        [
          1.937035350647955,
          1.7896880111358957
        ],
        [
          -0.7164206526733796,
          -1.6854606372237377
        ]
      ],
      "endpoint_formula": 2.2469437251375863,
      "independent_quadrature": 2.2469437251375863,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 2.4946086586847674e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 7,
      "first": [
        [
          1.6537927465851112,
          1.9341652310201294
        ],
        [
          0.710363534602724,
          -2.4748446955626955
        ]
      ],
      "second": [
        [
          -2.4377647291949054,
          1.1529979006308233
        ],
        [
          -2.100837466771831,
          0.10975817673732635
        ]
      ],
      "endpoint_formula": 0.9935051527302337,
      "independent_quadrature": 0.9935051527302327,
      "absolute_discrepancy": 9.992007221626409e-16,
      "quadrature_reported_error": 1.103012295644837e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 8,
      "first": [
        [
          2.870055486707666,
          -2.016640711332311
        ],
        [
          1.3111444296565544,
          2.360729525095077
        ]
      ],
      "second": [
        [
          -0.4712758883230679,
          -1.6437962729573543
        ],
        [
          2.237289406559843,
          1.0185490628156675
        ]
      ],
      "endpoint_formula": 4.4127317357808815,
      "independent_quadrature": 4.412731735780881,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 4.899116374558067e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 9,
      "first": [
        [
          1.187315693447843,
          -1.1907201673421697
        ],
        [
          -2.610496519258464,
          0.5785025440581251
        ]
      ],
      "second": [
        [
          -2.633387531202534,
          2.9363076025592214
        ],
        [
          -1.7228195096576409,
          2.4985832020178327
        ]
      ],
      "endpoint_formula": 0.6825752800404317,
      "independent_quadrature": 0.6825752800404317,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 7.578107919408513e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 10,
      "first": [
        [
          -2.303718993378788,
          -2.2343829644409245
        ],
        [
          -2.198372061764461,
          0.30796453161561743
        ]
      ],
      "second": [
        [
          -2.7376914393224245,
          1.0831969787993874
        ],
        [
          -1.9835086428071713,
          1.7616539911878633
        ]
      ],
      "endpoint_formula": 0.14823005943173895,
      "independent_quadrature": 0.14823005943173861,
      "absolute_discrepancy": 3.3306690738754696e-16,
      "quadrature_reported_error": 1.6456842492267157e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 11,
      "first": [
        [
          -2.2023149562262816,
          2.4521192871648614
        ],
        [
          -2.986532479966132,
          -1.3747173236771932
        ]
      ],
      "second": [
        [
          0.37213004117966086,
          0.5756456410950621
        ],
        [
          -2.6070128028684465,
          -1.9240595364195476
        ]
      ],
      "endpoint_formula": 4.425303831473445,
      "independent_quadrature": 4.425303831473445,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.9130742046637425e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 12,
      "first": [
        [
          -1.0923270682245108,
          -0.04948877634712945
        ],
        [
          -2.2749845388357337,
          1.6730794096800574
        ]
      ],
      "second": [
        [
          2.537148588321884,
          -1.740884143570464
        ],
        [
          2.1144487168351285,
          -2.4190481005634896
        ]
      ],
      "endpoint_formula": 0.1178816042822537,
      "independent_quadrature": 0.11788160428225308,
      "absolute_discrepancy": 6.106226635438361e-16,
      "quadrature_reported_error": 1.3087487125390883e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 13,
      "first": [
        [
          1.3879492754799303,
          -1.774566684012445
        ],
        [
          -2.223041481612781,
          -2.921954184283419
        ]
      ],
      "second": [
        [
          0.2924244708473438,
          1.5527373414536862
        ],
        [
          -0.730066312903614,
          -0.7776647460592456
        ]
      ],
      "endpoint_formula": 1.4071574936614337,
      "independent_quadrature": 1.407157493661435,
      "absolute_discrepancy": 1.3322676295501878e-15,
      "quadrature_reported_error": 1.562258648736753e-14,
      "symmetry_error": 2.6645352591003757e-15
    },
    {
      "index": 14,
      "first": [
        [
          2.084351340292728,
          1.829161313290502
        ],
        [
          -1.6527169109470865,
          -2.5856649895793202
        ]
      ],
      "second": [
        [
          -2.232264196971105,
          -1.5509112703914678
        ],
        [
          -1.2448176797074522,
          1.177689210381959
        ]
      ],
      "endpoint_formula": 4.689608514896562,
      "independent_quadrature": 4.689608514896562,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 5.2065113497163495e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 15,
      "first": [
        [
          1.4292240455711998,
          -0.9075925656676285
        ],
        [
          -1.1118664938376042,
          -0.8928415759938488
        ]
      ],
      "second": [
        [
          -0.26360919499834834,
          -0.8683050397921921
        ],
        [
          2.510078794965114,
          -1.1578805741644589
        ]
      ],
      "endpoint_formula": 3.376806655433738,
      "independent_quadrature": 3.3768066554337373,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 3.7490084985700025e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 16,
      "first": [
        [
          -1.223290592815104,
          -0.4760287011455051
        ],
        [
          0.8686183152303739,
          2.621606377462294
        ]
      ],
      "second": [
        [
          -0.1977138691458924,
          -1.6246788416686186
        ],
        [
          -0.2978281510101888,
          2.336788476004907
        ]
      ],
      "endpoint_formula": 5.321271178632153,
      "independent_quadrature": 5.321271178632152,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.907797782791659e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 17,
      "first": [
        [
          -2.351979830070422,
          -2.605146607472648
        ],
        [
          1.9243462001612945,
          -0.8837957278547881
        ]
      ],
      "second": [
        [
          0.3685500298860114,
          -0.5771853204752286
        ],
        [
          1.3691616601074035,
          0.6861310826325537
        ]
      ],
      "endpoint_formula": 1.5690580436625319,
      "independent_quadrature": 1.5690580436625308,
      "absolute_discrepancy": 1.1102230246251565e-15,
      "quadrature_reported_error": 1.742004367047446e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 18,
      "first": [
        [
          -2.905497103919463,
          1.8287990169551724
        ],
        [
          -0.13907869763435654,
          -2.3736346180629058
        ]
      ],
      "second": [
        [
          0.30384289879221527,
          -2.667792067713713
        ],
        [
          -1.0508040278691537,
          -0.20983949715977612
        ]
      ],
      "endpoint_formula": 4.5846744802196255,
      "independent_quadrature": 4.584674480219626,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.0900111683512004e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 19,
      "first": [
        [
          2.401450668607067,
          1.313212837592733
        ],
        [
          1.4138411840869534,
          0.6808094412836994
        ]
      ],
      "second": [
        [
          -0.6872435573742006,
          2.724200599310657
        ],
        [
          -1.9464470816214117,
          -0.9368347956257956
        ]
      ],
      "endpoint_formula": 0.70136870024221,
      "independent_quadrature": 0.7013687002422109,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 7.78675679760322e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 20,
      "first": [
        [
          0.5313017951294796,
          0.852266011385808
        ],
        [
          0.04198034349316648,
          1.9131739409148638
        ]
      ],
      "second": [
        [
          -1.4562545602176786,
          0.16997067546409728
        ],
        [
          2.9201704500670598,
          -2.8699815787327925
        ]
      ],
      "endpoint_formula": 0.9799895535520111,
      "independent_quadrature": 0.9799895535520098,
      "absolute_discrepancy": 1.3322676295501878e-15,
      "quadrature_reported_error": 1.0880069662455691e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 21,
      "first": [
        [
          -0.7123011876273528,
          -1.727594644620519
        ],
        [
          -0.731969137678862,
          -0.09836143207004167
        ]
      ],
      "second": [
        [
          -0.1707923794830446,
          -0.6485047084881943
        ],
        [
          -1.8742563336933582,
          2.751273890751996
        ]
      ],
      "endpoint_formula": 1.2801436871230092,
      "independent_quadrature": 1.280143687123009,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.421244996272507e-14,
      "symmetry_error": 6.661338147750939e-16
    },
    {
      "index": 22,
      "first": [
        [
          1.3258199691839945,
          -0.6301006857635594
        ],
        [
          -1.4250060328806673,
          -2.486453640696657
        ]
      ],
      "second": [
        [
          0.00043533431685460755,
          -1.0540444182003936
        ],
        [
          -2.4148021318987194,
          -2.7931542907742077
        ]
      ],
      "endpoint_formula": 3.9140523223231822,
      "independent_quadrature": 3.914052322323181,
      "absolute_discrepancy": 1.3322676295501878e-15,
      "quadrature_reported_error": 4.34547100783076e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 23,
      "first": [
        [
          0.8635830856434286,
          2.956670916739659
        ],
        [
          0.7697513464839032,
          -1.9471401629550997
        ]
      ],
      "second": [
        [
          2.7864867808846974,
          2.4770658775937893
        ],
        [
          2.7286113888999157,
          0.5939706441835053
        ]
      ],
      "endpoint_formula": 2.6819822672513416,
      "independent_quadrature": 2.681982267251342,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 2.97759846473882e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 24,
      "first": [
        [
          2.408138922795894,
          -2.4981011069669328
        ],
        [
          0.9181235871243225,
          -1.8758606948639527
        ]
      ],
      "second": [
        [
          -1.0913736428337082,
          -1.9199960716351163
        ],
        [
          -0.5834042360293328,
          2.763639631779454
        ]
      ],
      "endpoint_formula": 0.5162076748081224,
      "independent_quadrature": 0.5162076748081217,
      "absolute_discrepancy": 6.661338147750939e-16,
      "quadrature_reported_error": 5.731056460601921e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 25,
      "first": [
        [
          -0.6024460948100243,
          -2.1176655885120237
        ],
        [
          -1.7880426700292575,
          -1.8432345121776346
        ]
      ],
      "second": [
        [
          0.0993649252571478,
          -1.0783553514693998
        ],
        [
          -0.6173976771298362,
          1.6322852155216863
        ]
      ],
      "endpoint_formula": 0.8650331742522406,
      "independent_quadrature": 0.8650331742522401,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 9.603797471194221e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 26,
      "first": [
        [
          2.5883828930338515,
          2.029593944690747
        ],
        [
          -2.0904712719175973,
          0.8888677958778874
        ]
      ],
      "second": [
        [
          0.8376436717249773,
          -0.09634891397393108
        ],
        [
          -0.554959747588029,
          -2.822172623358475
        ]
      ],
      "endpoint_formula": 2.1681768657422062,
      "independent_quadrature": 2.1681768657422062,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 2.4071598778066042e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 27,
      "first": [
        [
          0.39447749826225076,
          1.6244386819792647
        ],
        [
          -2.4716259756949084,
          -2.375675177769677
        ]
      ],
      "second": [
        [
          2.0246325066649833,
          1.872813042405805
        ],
        [
          0.06677634255573928,
          -1.3460904639567939
        ]
      ],
      "endpoint_formula": 4.786275983730935,
      "independent_quadrature": 4.786275983730936,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.313833799348506e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 28,
      "first": [
        [
          2.196233073436928,
          -1.0141371426750159
        ],
        [
          -2.0743889640448945,
          2.807901107496212
        ]
      ],
      "second": [
        [
          2.699889238962503,
          -2.963958451846798
        ],
        [
          -1.5178774665142107,
          -0.5956497981593305
        ]
      ],
      "endpoint_formula": 5.765577841706117,
      "independent_quadrature": 5.765577841706116,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 6.401077270130746e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 29,
      "first": [
        [
          1.6121049378181542,
          2.6673830177630977
        ],
        [
          1.388738641130578,
          1.6245648054002153
        ]
      ],
      "second": [
        [
          -2.291029875964912,
          -2.9612945191648237
        ],
        [
          -1.0515997331366653,
          1.6641392140405245
        ]
      ],
      "endpoint_formula": 0.5641554493811878,
      "independent_quadrature": 0.5641554493811878,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 6.263383693707467e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 30,
      "first": [
        [
          -1.152704052708308,
          1.4578795862123215
        ],
        [
          0.23875823700514243,
          0.7126805042873388
        ]
      ],
      "second": [
        [
          2.556624479365663,
          0.3581384746675478
        ],
        [
          1.1272652460205954,
          -2.674539926776439
        ]
      ],
      "endpoint_formula": 0.41495328450987,
      "independent_quadrature": 0.4149532845098714,
      "absolute_discrepancy": 1.3877787807814457e-15,
      "quadrature_reported_error": 4.606906906066925e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 31,
      "first": [
        [
          2.914047386053949,
          -0.9808305869819662
        ],
        [
          -0.5547868022768396,
          2.8243734697625555
        ]
      ],
      "second": [
        [
          -1.2754156254381128,
          0.17979468683766608
        ],
        [
          -1.719145025648349,
          -1.7548481357437915
        ]
      ],
      "endpoint_formula": 1.633786965004008,
      "independent_quadrature": 1.6337869650040093,
      "absolute_discrepancy": 1.3322676295501878e-15,
      "quadrature_reported_error": 1.813867905879906e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 32,
      "first": [
        [
          1.9518703531341162,
          -2.6201915104657196
        ],
        [
          -1.9630198710730467,
          -0.47423870340744
        ]
      ],
      "second": [
        [
          0.11586687446147437,
          -1.652802212378369
        ],
        [
          -2.055482182922921,
          2.5609798870225386
        ]
      ],
      "endpoint_formula": 4.3579027011486815,
      "independent_quadrature": 4.3579027011486815,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.8382439178914285e-14,
      "symmetry_error": 1.7763568394002505e-15
    },
    {
      "index": 33,
      "first": [
        [
          2.3363358324388006,
          -1.0306211603411264
        ],
        [
          2.9615400794114475,
          -1.4031741100164816
        ]
      ],
      "second": [
        [
          1.0533725835513623,
          -0.3292606452396938
        ],
        [
          -2.7003133910619246,
          -1.4317586100045512
        ]
      ],
      "endpoint_formula": 0.11812632037160586,
      "independent_quadrature": 0.11812632037160564,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.3114656069080425e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 34,
      "first": [
        [
          -2.9082601856615398,
          2.396355112981145
        ],
        [
          -0.6303255006647959,
          -2.980763095624163
        ]
      ],
      "second": [
        [
          2.570104495900715,
          -2.556545656121295
        ],
        [
          2.549087601093243,
          -1.63286161667946
        ]
      ],
      "endpoint_formula": 0.826680792310218,
      "independent_quadrature": 0.8266807923102184,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 9.178000496381715e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 35,
      "first": [
        [
          -1.785740968994696,
          -2.600285980985719
        ],
        [
          -1.2343933863901788,
          -1.6331808551488909
        ]
      ],
      "second": [
        [
          2.6589268506287267,
          -0.23203028481596188
        ],
        [
          2.9567787269714207,
          2.065967646248012
        ]
      ],
      "endpoint_formula": 0.1646233516840141,
      "independent_quadrature": 0.16462335168401382,
      "absolute_discrepancy": 2.7755575615628914e-16,
      "quadrature_reported_error": 1.8276863543055667e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 36,
      "first": [
        [
          0.20858315295654517,
          0.8836754829097146
        ],
        [
          2.0983330941190967,
          2.3447640707354225
        ]
      ],
      "second": [
        [
          0.938751286637117,
          0.3972404492938182
        ],
        [
          2.1426114209613463,
          2.4728565704201007
        ]
      ],
      "endpoint_formula": 3.7936791170055004,
      "independent_quadrature": 3.793679117005501,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 4.21182990373914e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 37,
      "first": [
        [
          -0.743814187928499,
          -2.466055633890243
        ],
        [
          -0.8355991700941452,
          -0.4408110111910011
        ]
      ],
      "second": [
        [
          -1.1318916863257982,
          0.40470380751610957
        ],
        [
          -2.504776471905959,
          1.8992974608333935
        ]
      ],
      "endpoint_formula": 0.16820588052467111,
      "independent_quadrature": 0.16820588052467067,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 1.8674604143583757e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 38,
      "first": [
        [
          1.9478236658812467,
          1.9125471855117286
        ],
        [
          -1.8035861994012174,
          -2.6711237898098523
        ]
      ],
      "second": [
        [
          0.9990273733542838,
          0.831744814679896
        ],
        [
          -0.42212590026161934,
          0.922950764105297
        ]
      ],
      "endpoint_formula": 2.3536992177101244,
      "independent_quadrature": 2.353699217710124,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 2.6131310645439987e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 39,
      "first": [
        [
          -0.9777816876522571,
          0.8615203236173361
        ],
        [
          -2.1064422447030595,
          -1.9689554139353413
        ]
      ],
      "second": [
        [
          2.432375472987059,
          2.8039941171731186
        ],
        [
          -2.4149595243505386,
          -0.9944066529749804
        ]
      ],
      "endpoint_formula": 4.530362614438129,
      "independent_quadrature": 4.530362614438127,
      "absolute_discrepancy": 1.7763568394002505e-15,
      "quadrature_reported_error": 5.02971288445023e-14,
      "symmetry_error": 1.7763568394002505e-15
    },
    {
      "index": 40,
      "first": [
        [
          0.7165588997174659,
          -2.113717349069634
        ],
        [
          1.7340103587103473,
          -1.3630524230993293
        ]
      ],
      "second": [
        [
          2.0630869853108287,
          -1.85036089656909
        ],
        [
          -1.7273300394072133,
          2.16082698646931
        ]
      ],
      "endpoint_formula": 1.7429491869394202,
      "independent_quadrature": 1.7429491869394205,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.935062318091841e-14,
      "symmetry_error": 2.220446049250313e-16
    },
    {
      "index": 41,
      "first": [
        [
          2.969648197218996,
          -1.5658410751127032
        ],
        [
          -2.50781821227808,
          -0.722191544790924
        ]
      ],
      "second": [
        [
          -2.285760764293962,
          2.2043165446746853
        ],
        [
          1.7317859806288434,
          -2.2900871784737973
        ]
      ],
      "endpoint_formula": 7.783502786750057,
      "independent_quadrature": 7.783502786750056,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 8.641424006083982e-14,
      "symmetry_error": 1.7763568394002505e-15
    },
    {
      "index": 42,
      "first": [
        [
          -1.041453820521063,
          2.687581616297569
        ],
        [
          2.286957641541531,
          -2.8920307146328583
        ]
      ],
      "second": [
        [
          -0.8297065743968148,
          -0.1344328296282784
        ],
        [
          1.5636808275705025,
          -0.043584098552120754
        ]
      ],
      "endpoint_formula": 4.079471320153092,
      "independent_quadrature": 4.079471320153092,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.529122987931946e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 43,
      "first": [
        [
          0.3471006606824045,
          1.4413565790415994
        ],
        [
          2.3376520340561546,
          -0.39204406377977197
        ]
      ],
      "second": [
        [
          -2.2318709706135795,
          -2.4601545375724445
        ],
        [
          1.5912156411498142,
          0.7091739400968748
        ]
      ],
      "endpoint_formula": 2.87782856037223,
      "independent_quadrature": 2.87782856037223,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 3.195031528649117e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 44,
      "first": [
        [
          -1.6609449315868687,
          0.571651933663591
        ],
        [
          -2.2309313802864423,
          -1.2162030189996968
        ]
      ],
      "second": [
        [
          1.9923068333315506,
          -0.2583673760558769
        ],
        [
          -2.3399091681386057,
          2.424677651550631
        ]
      ],
      "endpoint_formula": 1.4998783306504782,
      "independent_quadrature": 1.4998783306504784,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.6651994568245046e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 45,
      "first": [
        [
          -1.886624784748076,
          -1.1459657590697638
        ],
        [
          -2.337381324150174,
          -2.892500518365133
        ]
      ],
      "second": [
        [
          -0.17651448447507923,
          1.7028844042973583
        ],
        [
          -2.706524860099714,
          -2.881591171153299
        ]
      ],
      "endpoint_formula": 3.3904000421339884,
      "independent_quadrature": 3.390400042133988,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 3.764100189467255e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 46,
      "first": [
        [
          1.495348989157625,
          -1.4138491510247144
        ],
        [
          -1.344167613365495,
          -0.8499741260806286
        ]
      ],
      "second": [
        [
          2.5174722973108956,
          2.44047019588145
        ],
        [
          -2.4676771461015643,
          -2.9886303742171574
        ]
      ],
      "endpoint_formula": 4.790085840590907,
      "independent_quadrature": 4.790085840590907,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 5.3180635901549717e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 47,
      "first": [
        [
          -1.6268621871956124,
          1.295120617908811
        ],
        [
          -1.6830883992252843,
          -2.7881683076984007
        ]
      ],
      "second": [
        [
          -1.5194074088574094,
          1.5646316959434774
        ],
        [
          1.3583709407746118,
          -1.2794117102865108
        ]
      ],
      "endpoint_formula": 4.612676045227528,
      "independent_quadrature": 4.612676045227529,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.1210991505485125e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 48,
      "first": [
        [
          -1.5268630306994175,
          1.1140086982275212
        ],
        [
          -0.6087420459862871,
          0.39313319790815315
        ]
      ],
      "second": [
        [
          1.1348916810979972,
          1.2718031618349457
        ],
        [
          -2.0061033376158965,
          -2.285244581257262
        ]
      ],
      "endpoint_formula": 1.2088154183679918,
      "independent_quadrature": 1.208815418367992,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.3420547099940361e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 49,
      "first": [
        [
          0.8387615846439034,
          -2.3638000252069173
        ],
        [
          1.3516624833810038,
          -2.939322831752697
        ]
      ],
      "second": [
        [
          0.39755183556068197,
          -1.0111739378718116
        ],
        [
          -1.8439764872810755,
          -0.4544445197392917
        ]
      ],
      "endpoint_formula": 0.05805312228706838,
      "independent_quadrature": 0.058053122287068694,
      "absolute_discrepancy": 3.122502256758253e-16,
      "quadrature_reported_error": 6.445191301448349e-16,
      "symmetry_error": 0.0
    },
    {
      "index": 50,
      "first": [
        [
          -0.5304766018877092,
          0.6287836870642689
        ],
        [
          1.6777159076057906,
          2.6647755033064886
        ]
      ],
      "second": [
        [
          1.3068112477090637,
          -0.5940751547703762
        ],
        [
          0.4675983754457147,
          0.6625118262086747
        ]
      ],
      "endpoint_formula": 0.8733687929113577,
      "independent_quadrature": 0.8733687929113573,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 9.696341428792691e-15,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 51,
      "first": [
        [
          -1.1317394513398742,
          -2.447735175964292
        ],
        [
          -2.459008242664636,
          -0.2513522635072949
        ]
      ],
      "second": [
        [
          1.8805202238418648,
          0.31836165053951415
        ],
        [
          -0.6096667089502485,
          -1.5085499754920917
        ]
      ],
      "endpoint_formula": 1.021290678104749,
      "independent_quadrature": 1.0212906781047488,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.1338604256669313e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 52,
      "first": [
        [
          1.1330319610009347,
          0.9572187484834376
        ],
        [
          1.4158933734926276,
          -1.308161805812597
        ]
      ],
      "second": [
        [
          -2.4931828811867645,
          -0.9020159012480016
        ],
        [
          -2.2064974451126,
          -2.002264895946402
        ]
      ],
      "endpoint_formula": 0.6289323521886181,
      "independent_quadrature": 0.6289323521886179,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 6.982551783314615e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 53,
      "first": [
        [
          -0.2493980982749342,
          -2.617882179579344
        ],
        [
          -1.7067725142375405,
          -2.1742458580517106
        ]
      ],
      "second": [
        [
          0.4081628362349843,
          -1.8791551764325924
        ],
        [
          2.437591332201597,
          0.9112619086079063
        ]
      ],
      "endpoint_formula": 0.41522858114658756,
      "independent_quadrature": 0.4152285811465871,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 4.609963312713763e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 54,
      "first": [
        [
          -0.3362383163995615,
          -1.360220667858155
        ],
        [
          -0.23887431410451976,
          1.7411090982706776
        ]
      ],
      "second": [
        [
          -0.7667227052874517,
          1.5881091500389024
        ],
        [
          -2.6660522758157104,
          -1.2840064720352489
        ]
      ],
      "endpoint_formula": 3.977424481212469,
      "independent_quadrature": 3.977424481212469,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.415828237749851e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 55,
      "first": [
        [
          2.7137443317969847,
          -1.0651780832502302
        ],
        [
          1.642004733047436,
          -0.8780515468315979
        ]
      ],
      "second": [
        [
          0.37582340077605414,
          0.4734871481932945
        ],
        [
          0.9387104982204408,
          0.006399400621864171
        ]
      ],
      "endpoint_formula": 0.015483304470237158,
      "independent_quadrature": 0.01548330447023667,
      "absolute_discrepancy": 4.874572967494828e-16,
      "quadrature_reported_error": 1.7189921120138365e-16,
      "symmetry_error": 0.0
    },
    {
      "index": 56,
      "first": [
        [
          -2.2730337832380174,
          -1.2722440344792967
        ],
        [
          1.4918805562896749,
          1.2118781719078706
        ]
      ],
      "second": [
        [
          -2.9499238038225686,
          0.8527671005368735
        ],
        [
          2.242182321726519,
          0.33105090836976725
        ]
      ],
      "endpoint_formula": 6.8226856795187905,
      "independent_quadrature": 6.8226856795187905,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 7.574702731182093e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 57,
      "first": [
        [
          -2.9625900107517937,
          1.4714475619740384
        ],
        [
          -1.2004079043926055,
          2.686573266769315
        ]
      ],
      "second": [
        [
          -0.18379329254812182,
          2.090156678670202
        ],
        [
          -1.42705698226732,
          -1.8319644524687695
        ]
      ],
      "endpoint_formula": 2.549555039734982,
      "independent_quadrature": 2.549555039734982,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 2.8305747076628828e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 58,
      "first": [
        [
          0.1762130348890234,
          -0.53467103129086
        ],
        [
          0.7680239105119719,
          -0.8626030731880396
        ]
      ],
      "second": [
        [
          -1.1313516975745952,
          1.742394020971223
        ],
        [
          -2.6507663137244277,
          1.7205579930414805
        ]
      ],
      "endpoint_formula": 0.07049141949965998,
      "independent_quadrature": 0.07049141949965937,
      "absolute_discrepancy": 6.106226635438361e-16,
      "quadrature_reported_error": 7.826119696703256e-16,
      "symmetry_error": 0.0
    },
    {
      "index": 59,
      "first": [
        [
          0.1276374505434985,
          -2.7911558987966254
        ],
        [
          2.166417273463642,
          -2.922298209044292
        ]
      ],
      "second": [
        [
          1.2753820763358128,
          -2.149696862684379
        ],
        [
          0.5841546427857782,
          1.0772526818714363
        ]
      ],
      "endpoint_formula": 0.8980711995708359,
      "independent_quadrature": 0.8980711995708363,
      "absolute_discrepancy": 3.3306690738754696e-16,
      "quadrature_reported_error": 9.970593235162764e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 60,
      "first": [
        [
          0.9022071770034739,
          -1.9478427147397819
        ],
        [
          0.8679269802042775,
          -1.2758144780200509
        ]
      ],
      "second": [
        [
          0.9919676291591877,
          -2.824098778042129
        ],
        [
          -2.8135804835562883,
          -2.1883543710768456
        ]
      ],
      "endpoint_formula": 0.604045709392575,
      "independent_quadrature": 0.604045709392575,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 6.706254544936729e-15,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 61,
      "first": [
        [
          0.21034488253175,
          -2.3687715868944528
        ],
        [
          -1.3300065433189168,
          0.8680158533822127
        ]
      ],
      "second": [
        [
          -2.2508590013880916,
          2.94945225379501
        ],
        [
          0.4867879021099366,
          2.65095165146965
        ]
      ],
      "endpoint_formula": 1.102261187631159,
      "independent_quadrature": 1.1022611876311588,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.2237557496587824e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 62,
      "first": [
        [
          -1.1934509014805303,
          2.0629144905783114
        ],
        [
          0.08347758532054517,
          -0.498843217249485
        ]
      ],
      "second": [
        [
          -1.8895229344109674,
          -2.5530728742772384
        ],
        [
          2.765702949484905,
          2.069949687357129
        ]
      ],
      "endpoint_formula": 3.659558384105666,
      "independent_quadrature": 3.659558384105665,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 4.062925977994142e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 63,
      "first": [
        [
          -0.7601061357135688,
          -2.5365656568808266
        ],
        [
          2.015523013646188,
          1.1928079550455353
        ]
      ],
      "second": [
        [
          0.33829853128558973,
          0.6806137153120035
        ],
        [
          -2.7763089210566907,
          0.043555887317014985
        ]
      ],
      "endpoint_formula": 3.3092088845981342,
      "independent_quadrature": 3.309208884598133,
      "absolute_discrepancy": 1.3322676295501878e-15,
      "quadrature_reported_error": 3.6739598969749796e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 64,
      "first": [
        [
          -0.5751271455545321,
          0.9457076644842868
        ],
        [
          -2.7670747238271067,
          -1.6894198987363611
        ]
      ],
      "second": [
        [
          1.2272286111084547,
          2.3552040728247468
        ],
        [
          1.5574586941415545,
          2.001475984541777
        ]
      ],
      "endpoint_formula": 0.09061162091603014,
      "independent_quadrature": 0.09061162091603074,
      "absolute_discrepancy": 5.967448757360216e-16,
      "quadrature_reported_error": 1.0059910783958374e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 65,
      "first": [
        [
          -1.0479976863365243,
          -0.8796295138011527
        ],
        [
          0.3658513858062946,
          1.5349643630749759
        ]
      ],
      "second": [
        [
          2.6063772038342856,
          -0.2606374522609478
        ],
        [
          0.10364248792709185,
          0.7893677122575049
        ]
      ],
      "endpoint_formula": 1.6685800410450362,
      "independent_quadrature": 1.6685800410450355,
      "absolute_discrepancy": 6.661338147750939e-16,
      "quadrature_reported_error": 1.8524959799981872e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 66,
      "first": [
        [
          1.6631508624988118,
          -1.9886588671174428
        ],
        [
          -0.4701670837680245,
          -0.8287349441161806
        ]
      ],
      "second": [
        [
          2.340604362778681,
          -2.3823921753041946
        ],
        [
          -2.1036424384168617,
          -2.010107384694332
        ]
      ],
      "endpoint_formula": 4.178971493690176,
      "independent_quadrature": 4.178971493690176,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.639590371547015e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 67,
      "first": [
        [
          2.5720120425480335,
          1.000131439104794
        ],
        [
          1.7763562485824362,
          2.0795025414159545
        ]
      ],
      "second": [
        [
          2.503916765423872,
          -2.0275395004655477
        ],
        [
          2.1042679236920634,
          -0.6861437443328584
        ]
      ],
      "endpoint_formula": 0.10747838956962874,
      "independent_quadrature": 0.10747838956962827,
      "absolute_discrepancy": 4.718447854656915e-16,
      "quadrature_reported_error": 1.1932498274983357e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 68,
      "first": [
        [
          -1.1888145952334255,
          -1.6598575095292951
        ],
        [
          2.7616772343837708,
          1.3460687612001
        ]
      ],
      "second": [
        [
          0.5024336151033264,
          -2.5476134934385537
        ],
        [
          1.90200998439368,
          -0.2070598397390957
        ]
      ],
      "endpoint_formula": 4.231634361412781,
      "independent_quadrature": 4.231634361412781,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.6980578998354407e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 69,
      "first": [
        [
          2.086034471662261,
          0.5866177349118082
        ],
        [
          -1.2697032145697158,
          0.12204142292052422
        ]
      ],
      "second": [
        [
          1.8225379007150142,
          -1.433622368895342
        ],
        [
          -1.372081881180997,
          0.9945712864208316
        ]
      ],
      "endpoint_formula": 4.941404482622092,
      "independent_quadrature": 4.941404482622091,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.4860610305930044e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 70,
      "first": [
        [
          0.8128231953583152,
          -1.682133021172577
        ],
        [
          2.0209732659161617,
          1.0130980186211715
        ]
      ],
      "second": [
        [
          -1.4279317822485256,
          -2.8292719657864875
        ],
        [
          -2.460395226512338,
          1.5200254357161418
        ]
      ],
      "endpoint_formula": 2.714999106138643,
      "independent_quadrature": 2.714999106138643,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 3.01425451947184e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 71,
      "first": [
        [
          2.1741980827315386,
          -0.4509553403047004
        ],
        [
          -0.5975133010875235,
          0.12338472539080847
        ]
      ],
      "second": [
        [
          -0.933086551641348,
          1.6065147950866665
        ],
        [
          -2.9475466924648224,
          -0.9407646125130906
        ]
      ],
      "endpoint_formula": 0.8897188331078667,
      "independent_quadrature": 0.8897188331078666,
      "absolute_discrepancy": 1.1102230246251565e-16,
      "quadrature_reported_error": 9.877863339589804e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 72,
      "first": [
        [
          -0.34287922347076094,
          -2.000120339475724
        ],
        [
          2.9733638763625505,
          -0.7975543422442302
        ]
      ],
      "second": [
        [
          -1.9366335573549514,
          -1.9499172137892098
        ],
        [
          -2.3572538767542124,
          1.589524939547501
        ]
      ],
      "endpoint_formula": 1.7243994208575253,
      "independent_quadrature": 1.724399420857525,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.9144679406863094e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 73,
      "first": [
        [
          0.9602935802691919,
          -2.3279402647814535
        ],
        [
          -2.198913383684406,
          2.846625622572949
        ]
      ],
      "second": [
        [
          1.080229595744738,
          -2.95057391331194
        ],
        [
          2.1963081964457096,
          -1.4449290850151078
        ]
      ],
      "endpoint_formula": 1.4753016928070257,
      "independent_quadrature": 1.475301692807026,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.6379139076228297e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 74,
      "first": [
        [
          -1.7215915528728045,
          -2.293780626689209
        ],
        [
          1.9722604058178064,
          1.7436920194944285
        ]
      ],
      "second": [
        [
          1.5448422648946973,
          1.6851828964109004
        ],
        [
          -2.216263349566873,
          0.39777145639557343
        ]
      ],
      "endpoint_formula": 6.379406585296727,
      "independent_quadrature": 6.379406585296727,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 7.082564074441775e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 75,
      "first": [
        [
          -0.12207961102438336,
          2.8508168828640725
        ],
        [
          2.046805587836479,
          2.3895763620428854
        ]
      ],
      "second": [
        [
          -1.9736435732419844,
          -2.712069149513499
        ],
        [
          -1.475307781193513,
          -1.2580939690981547
        ]
      ],
      "endpoint_formula": 0.11222211763058532,
      "independent_quadrature": 0.1122221176305871,
      "absolute_discrepancy": 1.790234627208065e-15,
      "quadrature_reported_error": 1.2459157886567053e-15,
      "symmetry_error": 1.7763568394002505e-15
    },
    {
      "index": 76,
      "first": [
        [
          -2.7124284973137205,
          1.6682648881067381
        ],
        [
          1.6095516701909052,
          -1.3004549579388591
        ]
      ],
      "second": [
        [
          -0.5935764786288398,
          -0.29972791787249475
        ],
        [
          -1.081380874520516,
          1.789050295728794
        ]
      ],
      "endpoint_formula": 3.732513919497854,
      "independent_quadrature": 3.732513919497854,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.1439228931604054e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 77,
      "first": [
        [
          -2.200610456262424,
          -2.4611815895821323
        ],
        [
          -1.5744624071379325,
          -1.6615264380281656
        ]
      ],
      "second": [
        [
          -1.6560616039002558,
          -2.5131204787918744
        ],
        [
          2.7699024012528994,
          0.8489878675233751
        ]
      ],
      "endpoint_formula": 1.2627555107394222,
      "independent_quadrature": 1.2627555107394222,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 1.4019402424952057e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 78,
      "first": [
        [
          2.593140359129265,
          1.4245081017181498
        ],
        [
          2.121333651802635,
          -1.7694413298277232
        ]
      ],
      "second": [
        [
          -2.43439153173583,
          2.4118995383641657
        ],
        [
          -2.5040988935126434,
          -1.2291899303411697
        ]
      ],
      "endpoint_formula": 2.1499045244177504,
      "independent_quadrature": 2.149904524417751,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 2.386873503754384e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 79,
      "first": [
        [
          -0.13123170318371535,
          -2.98471698253731
        ],
        [
          -2.5656294481943496,
          0.784754919229774
        ]
      ],
      "second": [
        [
          0.5299100211998415,
          2.4141986426463315
        ],
        [
          1.7182880223487693,
          -1.075920650185441
        ]
      ],
      "endpoint_formula": 3.9537306254751163,
      "independent_quadrature": 3.9537306254751163,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.3895227735680954e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 80,
      "first": [
        [
          -0.9115462724307708,
          2.8865336771291368
        ],
        [
          1.9412236532307832,
          -0.26928720047050403
        ]
      ],
      "second": [
        [
          -2.840408300090071,
          -1.7487972574589516
        ],
        [
          1.1490956879953877,
          1.9954657808104628
        ]
      ],
      "endpoint_formula": 4.780661908070584,
      "independent_quadrature": 4.7806619080705826,
      "absolute_discrepancy": 1.7763568394002505e-15,
      "quadrature_reported_error": 5.307600923288394e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 81,
      "first": [
        [
          -0.779105205854556,
          0.3709313457057948
        ],
        [
          -2.148955771441982,
          0.9578783133180604
        ]
      ],
      "second": [
        [
          0.39786949348550893,
          0.9525581789501656
        ],
        [
          1.6098866329741695,
          2.7726499852346613
        ]
      ],
      "endpoint_formula": 0.4475407293745177,
      "independent_quadrature": 0.4475407293745175,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 4.968700222091255e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 82,
      "first": [
        [
          2.9637129778577993,
          0.4840471105922184
        ],
        [
          1.04665203350479,
          0.8707321704915651
        ]
      ],
      "second": [
        [
          -0.7961400838783748,
          0.929236804420329
        ],
        [
          -1.4207054676717308,
          2.1914035079126712
        ]
      ],
      "endpoint_formula": 0.06239588874389246,
      "independent_quadrature": 0.062395888743892715,
      "absolute_discrepancy": 2.5673907444456745e-16,
      "quadrature_reported_error": 6.927335232541932e-16,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 83,
      "first": [
        [
          -1.7206730040939278,
          -0.6272233686867725
        ],
        [
          0.3013991747201681,
          -1.9924587076156146
        ]
      ],
      "second": [
        [
          -0.05091643670602597,
          -2.218988369068626
        ],
        [
          0.3911702861241029,
          -0.8244366518160007
        ]
      ],
      "endpoint_formula": 1.7872431641630082,
      "independent_quadrature": 1.7872431641630084,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 1.9842385114576903e-14,
      "symmetry_error": 2.220446049250313e-16
    },
    {
      "index": 84,
      "first": [
        [
          2.566586882377588,
          1.0400672869881298
        ],
        [
          -1.92123240172072,
          2.262886797944076
        ]
      ],
      "second": [
        [
          -0.9891864196352875,
          0.046247775732037866
        ],
        [
          0.20068639552292566,
          1.9046761603654838
        ]
      ],
      "endpoint_formula": 2.9487319063405577,
      "independent_quadrature": 2.948731906340558,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 3.273750055866118e-14,
      "symmetry_error": 1.3322676295501878e-15
    },
    {
      "index": 85,
      "first": [
        [
          1.0864692377261704,
          -1.5658774219649818
        ],
        [
          -0.03910865081521564,
          -0.8209872361371859
        ]
      ],
      "second": [
        [
          1.7164921856093542,
          2.4445432593912635
        ],
        [
          -0.6455310411581081,
          -1.3208238904720877
        ]
      ],
      "endpoint_formula": 1.384017046839943,
      "independent_quadrature": 1.3840170468399435,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 1.536567591875419e-14,
      "symmetry_error": 4.440892098500626e-16
    },
    {
      "index": 86,
      "first": [
        [
          1.8229308703107456,
          -1.5142123209217166
        ],
        [
          -2.3282884696517687,
          -0.6827032105008013
        ]
      ],
      "second": [
        [
          0.15243688932519817,
          -2.439625410601489
        ],
        [
          0.9896741891125274,
          -1.3598176198857772
        ]
      ],
      "endpoint_formula": 2.0169571154718895,
      "independent_quadrature": 2.0169571154718895,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 2.239272229278432e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 87,
      "first": [
        [
          -0.0065449914799469155,
          1.3803264194960772
        ],
        [
          1.9224691571461747,
          -2.2239674079990714
        ]
      ],
      "second": [
        [
          -1.140244476747789,
          -2.0938105473823985
        ],
        [
          2.073159090319196,
          0.6341038224495819
        ]
      ],
      "endpoint_formula": 4.8148837085966845,
      "independent_quadrature": 4.814883708596684,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.345594754176601e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 88,
      "first": [
        [
          -1.9509996655633985,
          2.9233840947862655
        ],
        [
          -0.8782931157186651,
          -1.5799460637814484
        ]
      ],
      "second": [
        [
          -2.189721279428404,
          -2.7741175767619994
        ],
        [
          1.846806319329838,
          -1.1686382634183816
        ]
      ],
      "endpoint_formula": 2.5029576491535783,
      "independent_quadrature": 2.5029576491535797,
      "absolute_discrepancy": 1.3322676295501878e-15,
      "quadrature_reported_error": 2.7788412117519587e-14,
      "symmetry_error": 1.3322676295501878e-15
    },
    {
      "index": 89,
      "first": [
        [
          -2.798681271229513,
          1.0921068557398819
        ],
        [
          -1.8431661197853144,
          2.335897858916665
        ]
      ],
      "second": [
        [
          -0.0016671015860163507,
          -0.651862972704766
        ],
        [
          2.0281335665010483,
          0.7514735625267299
        ]
      ],
      "endpoint_formula": 0.8693443445684834,
      "independent_quadrature": 0.8693443445684832,
      "absolute_discrepancy": 2.220446049250313e-16,
      "quadrature_reported_error": 9.651661076675957e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 90,
      "first": [
        [
          0.3764422847298947,
          -1.958386139200895
        ],
        [
          -0.9198357028793782,
          -1.4010503010387778
        ]
      ],
      "second": [
        [
          1.935073067032934,
          1.6957575573068127
        ],
        [
          1.6109925815038881,
          -0.8169013484388645
        ]
      ],
      "endpoint_formula": 0.6766643028215138,
      "independent_quadrature": 0.676664302821513,
      "absolute_discrepancy": 7.771561172376096e-16,
      "quadrature_reported_error": 7.512482889343731e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 91,
      "first": [
        [
          2.7442982470120114,
          1.2529552779837712
        ],
        [
          -2.516683589457495,
          -1.7426167067465075
        ]
      ],
      "second": [
        [
          -2.4678822055355707,
          -1.2182091381590592
        ],
        [
          -0.05868162859280357,
          -0.9999274831547567
        ]
      ],
      "endpoint_formula": 4.487336791120935,
      "independent_quadrature": 4.487336791120935,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 4.981944624750029e-14,
      "symmetry_error": 8.881784197001252e-16
    },
    {
      "index": 92,
      "first": [
        [
          -2.546979625870205,
          1.965283122478633
        ],
        [
          2.471894614647243,
          0.5722955567346979
        ]
      ],
      "second": [
        [
          1.6619439297967284,
          2.4431998386812035
        ],
        [
          -2.238643394247631,
          0.8026402060662825
        ]
      ],
      "endpoint_formula": 6.686628369855334,
      "independent_quadrature": 6.686628369855334,
      "absolute_discrepancy": 0.0,
      "quadrature_reported_error": 7.423648773325168e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 93,
      "first": [
        [
          -0.8729073090544786,
          2.2744741350927855
        ],
        [
          2.142083882279181,
          -0.5531973513353243
        ]
      ],
      "second": [
        [
          2.880040426349291,
          0.7518288095967698
        ],
        [
          -2.399683621035395,
          -1.9684294149038557
        ]
      ],
      "endpoint_formula": 5.329925867973787,
      "independent_quadrature": 5.329925867973788,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 5.917406418169722e-14,
      "symmetry_error": 1.7763568394002505e-15
    },
    {
      "index": 94,
      "first": [
        [
          -1.9105840263421296,
          -2.4303730696229096
        ],
        [
          -1.9673350382122747,
          0.7042079365319465
        ]
      ],
      "second": [
        [
          -1.0031607705369754,
          2.785882544773031
        ],
        [
          2.3633408698872636,
          -1.5431747871514956
        ]
      ],
      "endpoint_formula": 3.514520634762982,
      "independent_quadrature": 3.514520634762981,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 3.9019017292340815e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 95,
      "first": [
        [
          -1.7533327135492882,
          1.8119340527338954
        ],
        [
          -1.6738352409471993,
          0.5827489345964967
        ]
      ],
      "second": [
        [
          1.5273860508905006,
          -0.8677717977363173
        ],
        [
          0.21229553796737122,
          0.8099810565305376
        ]
      ],
      "endpoint_formula": 0.41501654114648057,
      "independent_quadrature": 0.41501654114648073,
      "absolute_discrepancy": 1.6653345369377348e-16,
      "quadrature_reported_error": 4.607609195811165e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 96,
      "first": [
        [
          2.4499515848574687,
          -2.0640609047965146
        ],
        [
          2.0981810753206496,
          2.2801222168108692
        ]
      ],
      "second": [
        [
          2.88538976598963,
          -0.942241565401214
        ],
        [
          1.5894520160145253,
          -0.541850183349605
        ]
      ],
      "endpoint_formula": 2.293018731052409,
      "independent_quadrature": 2.2930187310524093,
      "absolute_discrepancy": 4.440892098500626e-16,
      "quadrature_reported_error": 2.5457621911111442e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 97,
      "first": [
        [
          -1.8674010299984494,
          -0.2532890430009358
        ],
        [
          -0.5960893171501747,
          -0.11653762579248284
        ]
      ],
      "second": [
        [
          -0.6535952938548082,
          -1.5513201003945218
        ],
        [
          -2.663795931804132,
          0.11060460022058827
        ]
      ],
      "endpoint_formula": 1.693939644886691,
      "independent_quadrature": 1.6939396448866904,
      "absolute_discrepancy": 6.661338147750939e-16,
      "quadrature_reported_error": 1.8806507960785647e-14,
      "symmetry_error": 0.0
    },
    {
      "index": 98,
      "first": [
        [
          -0.5289576033428021,
          0.10116034812409813
        ],
        [
          0.08110301861771818,
          -2.8662258951224464
        ]
      ],
      "second": [
        [
          2.2754236053409675,
          -2.607054035401857
        ],
        [
          1.1097487751199955,
          -2.5089844051712937
        ]
      ],
      "endpoint_formula": 0.4022572482391569,
      "independent_quadrature": 0.4022572482391566,
      "absolute_discrepancy": 2.7755575615628914e-16,
      "quadrature_reported_error": 4.465952588174688e-15,
      "symmetry_error": 0.0
    },
    {
      "index": 99,
      "first": [
        [
          -2.5211163410756514,
          -2.2249819607105987
        ],
        [
          -1.8723205646245802,
          0.4558074200080857
        ]
      ],
      "second": [
        [
          -1.8761098942698007,
          -1.275159384925298
        ],
        [
          2.302797979428666,
          -1.3705664542362388
        ]
      ],
      "endpoint_formula": 1.7829839949543616,
      "independent_quadrature": 1.7829839949543624,
      "absolute_discrepancy": 8.881784197001252e-16,
      "quadrature_reported_error": 1.979509883736477e-14,
      "symmetry_error": 0.0
    }
  ],
  "actual_opaque_examples": {
    "square_boundary": {
      "segment_count": 4,
      "nondegenerate_segment_count": 4,
      "total_segment_length": 8.0,
      "pair_overlaps": [
        {
          "i": 0,
          "j": 1,
          "Q": 1.1715728752538097
        },
        {
          "i": 0,
          "j": 2,
          "Q": 1.6568542494923806
        },
        {
          "i": 0,
          "j": 3,
          "Q": 1.1715728752538097
        },
        {
          "i": 1,
          "j": 2,
          "Q": 1.1715728752538097
        },
        {
          "i": 1,
          "j": 3,
          "Q": 1.6568542494923806
        },
        {
          "i": 2,
          "j": 3,
          "Q": 1.1715728752538097
        }
      ],
      "pair_energy": 8.0,
      "maximum_spanning_forest_edges": [
        {
          "i": 0,
          "j": 2,
          "Q": 1.6568542494923806
        },
        {
          "i": 1,
          "j": 3,
          "Q": 1.6568542494923806
        },
        {
          "i": 0,
          "j": 1,
          "Q": 1.1715728752538097
        }
      ],
      "maximum_spanning_forest_value": 4.485281374238571,
      "forest_necessary_lower_bound": 5.384233340709079,
      "automatic_hit_bound": 4,
      "automatic_pair_necessary_lower_bound": 5.141592653589793,
      "opacity_established": false,
      "arithmetic": "float64 evaluation of exact geometric formulas",
      "externally_assumed_max_hits": 2,
      "conditional_pair_necessary_lower_bound": 7.141592653589793,
      "max_hits_assumption_verified_by_this_code": false,
      "independent_geometric_facts": "The square boundary surrounds the unit disk, so every line meeting the disk meets the boundary. A line not containing an edge and not through a vertex meets a convex polygon boundary at most twice; the exceptional lines have zero line measure."
    },
    "cross": {
      "segment_count": 2,
      "nondegenerate_segment_count": 2,
      "total_segment_length": 5.656854249492381,
      "pair_overlaps": [
        {
          "i": 0,
          "j": 1,
          "Q": 3.313708498984761
        }
      ],
      "pair_energy": 3.313708498984761,
      "maximum_spanning_forest_edges": [
        {
          "i": 0,
          "j": 1,
          "Q": 3.313708498984761
        }
      ],
      "maximum_spanning_forest_value": 3.313708498984761,
      "forest_necessary_lower_bound": 4.798446903082174,
      "automatic_hit_bound": 2,
      "automatic_pair_necessary_lower_bound": 4.798446903082174,
      "opacity_established": false,
      "arithmetic": "float64 evaluation of exact geometric formulas",
      "externally_assumed_max_hits": 2,
      "conditional_pair_necessary_lower_bound": 4.798446903082174,
      "max_hits_assumption_verified_by_this_code": false,
      "independent_geometric_facts": "The cross is connected. Its convex hull is the diamond |x|+|y|<=sqrt(2), which contains the unit disk. Every linear projection of a connected set is an interval, so containment of the disk in its convex hull makes it opaque. Two segments give the automatic almost-everywhere hit bound 2."
    },
    "triangle_boundary": {
      "segment_count": 3,
      "nondegenerate_segment_count": 3,
      "total_segment_length": 10.392304845413264,
      "pair_overlaps": [
        {
          "i": 0,
          "j": 1,
          "Q": 3.4641016151377544
        },
        {
          "i": 0,
          "j": 2,
          "Q": 3.4641016151377544
        },
        {
          "i": 1,
          "j": 2,
          "Q": 3.4641016151377544
        }
      ],
      "pair_energy": 10.392304845413264,
      "maximum_spanning_forest_edges": [
        {
          "i": 0,
          "j": 1,
          "Q": 3.4641016151377544
        },
        {
          "i": 0,
          "j": 2,
          "Q": 3.4641016151377544
        }
      ],
      "maximum_spanning_forest_value": 6.928203230275509,
      "forest_necessary_lower_bound": 6.6056942687275475,
      "automatic_hit_bound": 3,
      "automatic_pair_necessary_lower_bound": 6.605694268727548,
      "opacity_established": false,
      "arithmetic": "float64 evaluation of exact geometric formulas",
      "externally_assumed_max_hits": 2,
      "conditional_pair_necessary_lower_bound": 8.337745076296425,
      "max_hits_assumption_verified_by_this_code": false,
      "independent_geometric_facts": "The equilateral triangle is circumscribed about the unit disk; its boundary is opaque and a generic line hits it at most twice."
    }
  },
  "fractional_no_go": {
    "analytic_statement": "Half of the unit-circle boundary measure has cost pi and weighted intersection count 1 on every non-tangent line through the disk. Consequently line-count covering constraints allowing fractional density cannot imply a bound strictly greater than pi. For a regular circumscribed N-gon, half boundary density has cost N*tan(pi/N), tending to pi. This is a fractional relaxation, not a physical barrier.",
    "polygon_diagnostics": [
      {
        "circumscribed_regular_polygon_sides": 3,
        "half_density_cost": 5.19615242270663,
        "excess_above_pi": 2.054559769116837
      },
      {
        "circumscribed_regular_polygon_sides": 4,
        "half_density_cost": 3.9999999999999996,
        "excess_above_pi": 0.8584073464102064
      },
      {
        "circumscribed_regular_polygon_sides": 6,
        "half_density_cost": 3.4641016151377544,
        "excess_above_pi": 0.32250896154796127
      },
      {
        "circumscribed_regular_polygon_sides": 12,
        "half_density_cost": 3.2153903091734723,
        "excess_above_pi": 0.07379765558367923
      },
      {
        "circumscribed_regular_polygon_sides": 24,
        "half_density_cost": 3.1596599420975,
        "excess_above_pi": 0.01806728850770689
      },
      {
        "circumscribed_regular_polygon_sides": 96,
        "half_density_cost": 3.1427145996453683,
        "excess_above_pi": 0.0011219460555751404
      },
      {
        "circumscribed_regular_polygon_sides": 384,
        "half_density_cost": 3.141662747056848,
        "excess_above_pi": 7.009346705499908e-05
      }
    ]
  }
}