Hugging Face's logo

Datasets:
PureOne
/
neoryae-aegis

Languages:
English
Tags:
math
geometry
opaque-barriers
reproducible-research
ai-assisted-research
exact-arithmetic
Dataset card Files
xet
 Community
neoryae-aegis / POSITIONAL_VERIFICATION.json
PureOne's picture
PureOne
Publish Neoryae Aegis research artifacts v1.0.0
3b80460 verified
Raw History Blame Contribute Delete
72.6 kB
{
"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
}
]
}
}