{ "version": "2.0.0-junctions", "method": "integer-dyadic arithmetic; rational Machin and Taylor bounds", "dyadic_bits": 100, "count": 23, "all_passed": true, "scope": "Junction radii, dimensions and contact diameters, m=2,...,24. Not a global contact-atlas classification.", "seams": [ { "m": 2, "theta_over_pi": "1/4", "theta_degrees": 45.0, "R_isolating_bracket": [ "1018531079741487693907243/604462909807314587353088", "10185310797414876939072433/6044629098073145873530880" ], "R": { "lower": "1018531079741487693907243/604462909807314587353088", "upper": "1068007245471010200126481550541/633825300114114700748351602688", "approx": 1.685018324889721 }, "r": { "lower": "752306714712549540352347726723/1267650600228229401496703205376", "upper": "376153357356274770176173974155/633825300114114700748351602688", "approx": 0.5934653559719874 }, "p": { "lower": "446467972246999516226093918829/1267650600228229401496703205376", "upper": "223233986123499758113047090919/633825300114114700748351602688", "approx": 0.3522011287389576 }, "s": { "lower": "1063922359010834402002673200963/1267650600228229401496703205376", "upper": "1063922359010834402002673514337/1267650600228229401496703205376", "approx": 0.8392867552141612 }, "contact_left": { "lower": "2331572959239063803499376406339/1267650600228229401496703205376", "upper": "2331572959239063803499376719713/1267650600228229401496703205376", "approx": 1.8392867552141612 }, "contact_right": "2", "contact_diameter": { "lower": "203728241217394999494029691039/1267650600228229401496703205376", "upper": "203728241217394999494030004413/1267650600228229401496703205376", "approx": 0.16071324478583887 }, "contact_dimension": { "lower": "420998522414120331842251793335/633825300114114700748351602688", "upper": "841997044828240663684504387995/1267650600228229401496703205376", "approx": 0.6642185509766227 }, "canopy_dimension": { "lower": "420998522414120331842251793335/316912650057057350374175801344", "upper": "841997044828240663684504387995/633825300114114700748351602688", "approx": 1.3284371019532455 }, "radius_excess_over_2cos": { "lower": "21455363734303995178346278041/79228162514264337593543950336", "upper": "85821454937215980713385269451/316912650057057350374175801344", "approx": 0.2708047625166259 }, "normalized_radial_derivative": { "lower": "431748398994583695688538404659/316912650057057350374175801344", "upper": "863496797989167391377077670583/633825300114114700748351602688", "approx": 1.3623577314343596 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 3, "theta_over_pi": "1/6", "theta_degrees": 30.0, "R_isolating_bracket": [ "5461005027688907866413737/3022314549036572936765440", "10922010055377815732827477/6044629098073145873530880" ], "R": { "lower": "572627880791392425493265068851/316912650057057350374175801344", "upper": "2290511523165569701973060904551/1267650600228229401496703205376", "approx": 1.8068949935835499 }, "r": { "lower": "701562960066729472344320210187/1267650600228229401496703205376", "upper": "350781480033364736172160201445/633825300114114700748351602688", "approx": 0.5534355917477728 }, "p": { "lower": "388269911952849501790568459729/1267650600228229401496703205376", "upper": "97067477988212375447642168257/316912650057057350374175801344", "approx": 0.3062909542132074 }, "s": { "lower": "1215142691543990794386528980133/1267650600228229401496703205376", "upper": "1215142691543990794386529313907/1267650600228229401496703205376", "approx": 0.9585785636240893 }, "contact_left": { "lower": "2482793291772220195883232185509/1267650600228229401496703205376", "upper": "2482793291772220195883232519283/1267650600228229401496703205376", "approx": 1.9585785636240893 }, "contact_right": "2", "contact_diameter": { "lower": "52507908684238607110173891469/1267650600228229401496703205376", "upper": "52507908684238607110174225243/1267650600228229401496703205376", "approx": 0.0414214363759107 }, "contact_dimension": { "lower": "371303979595775848794830281703/633825300114114700748351602688", "upper": "92825994898943962198707648343/158456325028528675187087900672", "approx": 0.5858143869121757 }, "canopy_dimension": { "lower": "371303979595775848794830281703/316912650057057350374175801344", "upper": "92825994898943962198707648343/79228162514264337593543950336", "approx": 1.1716287738243514 }, "radius_excess_over_2cos": { "lower": "23719069331273235083322621297/316912650057057350374175801344", "upper": "94876277325092940333291114337/1267650600228229401496703205376", "approx": 0.07484418601467258 }, "normalized_radial_derivative": { "lower": "870370317526109978520214200297/633825300114114700748351602688", "upper": "217592579381527494630053768673/158456325028528675187087900672", "approx": 1.3732022331222933 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 4, "theta_over_pi": "1/8", "theta_degrees": 22.5, "R_isolating_bracket": [ "11319118948431483670069111/6044629098073145873530880", "5659559474215741835034557/3022314549036572936765440" ], "R": { "lower": "2373791294094098284165277627187/1267650600228229401496703205376", "upper": "2373791294094098284165278256333/1267650600228229401496703205376", "approx": 1.8725911490648275 }, "r": { "lower": "338475005839180241128932898823/633825300114114700748351602688", "upper": "84618751459795060282233247133/158456325028528675187087900672", "approx": 0.5340193990019659 }, "p": { "lower": "361504438390851904855077660645/1267650600228229401496703205376", "upper": "361504438390851904855077852271/1267650600228229401496703205376", "approx": 0.2851767185104209 }, "s": { "lower": "1250840520645827584815312028757/1267650600228229401496703205376", "upper": "1250840520645827584815312360281/1267650600228229401496703205376", "approx": 0.9867391854037892 }, "contact_left": { "lower": "2518491120874056986312015234133/1267650600228229401496703205376", "upper": "2518491120874056986312015565657/1267650600228229401496703205376", "approx": 1.9867391854037892 }, "contact_right": "2", "contact_diameter": { "lower": "16810079582401816681390845095/1267650600228229401496703205376", "upper": "16810079582401816681391176619/1267650600228229401496703205376", "approx": 0.013260814596210745 }, "contact_dimension": { "lower": "350165816198576633420404014293/633825300114114700748351602688", "upper": "175082908099288316710202145745/316912650057057350374175801344", "approx": 0.5524642454877272 }, "canopy_dimension": { "lower": "350165816198576633420404014293/316912650057057350374175801344", "upper": "175082908099288316710202145745/158456325028528675187087900672", "approx": 1.1049284909754544 }, "radius_excess_over_2cos": { "lower": "31478406241081012289177578749/1267650600228229401496703205376", "upper": "31478406241081012289178207897/1267650600228229401496703205376", "approx": 0.024832084042253916 }, "normalized_radial_derivative": { "lower": "883804817532166597098724213087/633825300114114700748351602688", "upper": "883804817532166597098725118829/633825300114114700748351602688", "approx": 1.3943981367942322 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 5, "theta_over_pi": "1/10", "theta_degrees": 18.0, "R_isolating_bracket": [ "11552785807295728421170267/6044629098073145873530880", "1155278580729572842117027/604462909807314587353088" ], "R": { "lower": "1211397393067092572495703388979/633825300114114700748351602688", "upper": "1155278580729572842117027/604462909807314587353088", "approx": 1.911248088154561 }, "r": { "lower": "20726812757930467056714915197/39614081257132168796771975168", "upper": "663258008253774945814877458539/1267650600228229401496703205376", "approx": 0.5232183127861583 }, "p": { "lower": "347028736020447932122040863343/1267650600228229401496703205376", "upper": "347028736020447932122041043577/1267650600228229401496703205376", "approx": 0.2737574028347941 }, "s": { "lower": "39424740670918588386607653185/39614081257132168796771975168", "upper": "1261591701469394828371445229533/1267650600228229401496703205376", "approx": 0.9952203716404633 }, "contact_left": { "lower": "79038821928050757183379628353/39614081257132168796771975168", "upper": "2529242301697624229868148434909/1267650600228229401496703205376", "approx": 1.9952203716404633 }, "contact_right": "2", "contact_diameter": { "lower": "6058898758834573125257975843/1267650600228229401496703205376", "upper": "189340586213580410164321983/39614081257132168796771975168", "approx": 0.004779628359536706 }, "contact_dimension": { "lower": "84779974168044656472944947151/158456325028528675187087900672", "upper": "339119896672178625891780048589/633825300114114700748351602688", "approx": 0.5350368573345416 }, "canopy_dimension": { "lower": "84779974168044656472944947151/79228162514264337593543950336", "upper": "339119896672178625891780048589/316912650057057350374175801344", "approx": 1.0700737146690833 }, "radius_excess_over_2cos": { "lower": "1447507333643077551643334763/158456325028528675187087900672", "upper": "2895014667286155103286826813/316912650057057350374175801344", "approx": 0.00913505556425385 }, "normalized_radial_derivative": { "lower": "896717171656829719391963160339/633825300114114700748351602688", "upper": "448358585828414859695982049305/316912650057057350374175801344", "approx": 1.4147702395208643 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 6, "theta_over_pi": "1/12", "theta_degrees": 15.0, "R_isolating_bracket": [ "11699005361481815362758679/6044629098073145873530880", "5849502680740907681379341/3022314549036572936765440" ], "R": { "lower": "613364812296057801291002229555/316912650057057350374175801344", "upper": "2453459249184231205164009547367/1267650600228229401496703205376", "approx": 1.9354380842343366 }, "r": { "lower": "327484152180903958328830267997/633825300114114700748351602688", "upper": "654968304361807916657660703951/1267650600228229401496703205376", "approx": 0.516678889469927 }, "p": { "lower": "169204148067830018909164212009/633825300114114700748351602688", "upper": "338408296135660037818328597579/1267650600228229401496703205376", "approx": 0.266957074823877 }, "s": { "lower": "158162700145957320092035520755/158456325028528675187087900672", "upper": "632650800583829280368142245255/633825300114114700748351602688", "approx": 0.9981469664747148 }, "contact_left": { "lower": "316619025174485995279123421427/158456325028528675187087900672", "upper": "1266476100697943981116493847943/633825300114114700748351602688", "approx": 1.9981469664747147 }, "contact_right": "2", "contact_diameter": { "lower": "1174499530285420380209357433/633825300114114700748351602688", "upper": "293624882571355095052379917/158456325028528675187087900672", "approx": 0.0018530335252852199 }, "contact_dimension": { "lower": "665321517122079336277479598861/1267650600228229401496703205376", "upper": "665321517122079336277480099217/1267650600228229401496703205376", "approx": 0.5248461342599404 }, "canopy_dimension": { "lower": "665321517122079336277479598861/633825300114114700748351602688", "upper": "665321517122079336277480099217/633825300114114700748351602688", "approx": 1.0496922685198808 }, "radius_excess_over_2cos": { "lower": "2273171120829742218557880121/633825300114114700748351602688", "upper": "4546342241659484437116389391/1267650600228229401496703205376", "approx": 0.003586431656200025 }, "normalized_radial_derivative": { "lower": "907490627114442826328478988287/633825300114114700748351602688", "upper": "907490627114442826328479955395/633825300114114700748351602688", "approx": 1.4317677551701662 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 7, "theta_over_pi": "1/14", "theta_degrees": 12.857142857142858, "R_isolating_bracket": [ "11795049566305605110571403/6044629098073145873530880", "5897524783152802555285703/3022314549036572936765440" ], "R": { "lower": "2473601178807693236884503894425/1267650600228229401496703205376", "upper": "618400294701923309221126130893/316912650057057350374175801344", "approx": 1.9513272650699989 }, "r": { "lower": "5075273887452077019280918313/9903520314283042199192993792", "upper": "40602191099616616154247356831/79228162514264337593543950336", "approx": 0.5124716995968012 }, "p": { "lower": "83229895520698561184714272813/316912650057057350374175801344", "upper": "332919582082794244738857260607/1267650600228229401496703205376", "approx": 0.262627242887634 }, "s": { "lower": "633347350380106374821401775725/633825300114114700748351602688", "upper": "1266694700760212749642803873631/1267650600228229401496703205376", "approx": 0.9992459282803601 }, "contact_left": { "lower": "1267172650494221075569753378413/633825300114114700748351602688", "upper": "2534345300988442151139507079007/1267650600228229401496703205376", "approx": 1.99924592828036 }, "contact_right": "2", "contact_diameter": { "lower": "955899468016651853899331745/1267650600228229401496703205376", "upper": "477949734008325926949826963/633825300114114700748351602688", "approx": 0.0007540717196398996 }, "contact_dimension": { "lower": "328592210951442217010720124425/633825300114114700748351602688", "upper": "657184421902884434021440737043/1267650600228229401496703205376", "approx": 0.5184270979594567 }, "canopy_dimension": { "lower": "328592210951442217010720124425/316912650057057350374175801344", "upper": "657184421902884434021440737043/633825300114114700748351602688", "approx": 1.0368541959189135 }, "radius_excess_over_2cos": { "lower": "1865272694606800023730168955/1267650600228229401496703205376", "upper": "233159086825850002966349763/158456325028528675187087900672", "approx": 0.001471440706351556 }, "normalized_radial_derivative": { "lower": "916032069335383109829476386807/633825300114114700748351602688", "upper": "229008017333845777457369344489/158456325028528675187087900672", "approx": 1.445243774854774 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 8, "theta_over_pi": "1/16", "theta_degrees": 11.25, "R_isolating_bracket": [ "2965183002188914187560351/1511157274518286468382720", "11860732008755656750241407/6044629098073145873530880" ], "R": { "lower": "621843946340648576627056522035/316912650057057350374175801344", "upper": "2487375785362594306508226717287/1267650600228229401496703205376", "approx": 1.9621935136659614 }, "r": { "lower": "646037504150077892129338615081/1267650600228229401496703205376", "upper": "646037504150077892129338778489/1267650600228229401496703205376", "approx": 0.5096337303305536 }, "p": { "lower": "164621251586722354823012748923/633825300114114700748351602688", "upper": "329242503173444709646025664403/1267650600228229401496703205376", "approx": 0.2597265390906354 }, "s": { "lower": "633624074658837300404676898535/633825300114114700748351602688", "upper": "158406018664709325101169264701/158456325028528675187087900672", "approx": 0.9996825222103927 }, "contact_left": { "lower": "1267449374772952001153028501223/633825300114114700748351602688", "upper": "316862343693238000288257165373/158456325028528675187087900672", "approx": 1.9996825222103927 }, "contact_right": "2", "contact_diameter": { "lower": "50306363819350085918635971/158456325028528675187087900672", "upper": "201225455277400343674704153/633825300114114700748351602688", "approx": 0.00031747778960728053 }, "contact_dimension": { "lower": "40735642305577966117434386271/79228162514264337593543950336", "upper": "651770276889247457878950660523/1267650600228229401496703205376", "approx": 0.5141560906230013 }, "canopy_dimension": { "lower": "40735642305577966117434386271/39614081257132168796771975168", "upper": "651770276889247457878950660523/633825300114114700748351602688", "approx": 1.0283121812460025 }, "radius_excess_over_2cos": { "lower": "197421641564897468916570147/316912650057057350374175801344", "upper": "789686566259589875666909737/1267650600228229401496703205376", "approx": 0.0006229528595004125 }, "normalized_radial_derivative": { "lower": "230667333648719810741109347861/158456325028528675187087900672", "upper": "922669334594879242964438402077/633825300114114700748351602688", "approx": 1.45571553301637 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 9, "theta_over_pi": "1/18", "theta_degrees": 10.0, "R_isolating_bracket": [ "11907226902281115234082063/6044629098073145873530880", "5953613451140557617041033/3022314549036572936765440" ], "R": { "lower": "2497126471257264537538566658457/1267650600228229401496703205376", "upper": "624281617814316134384641821901/316912650057057350374175801344", "approx": 1.9698854485673567 }, "r": { "lower": "643514881106491108617507641461/1267650600228229401496703205376", "upper": "321757440553245554308753901797/633825300114114700748351602688", "approx": 0.5076437316328584 }, "p": { "lower": "81669073901543582230208012937/316912650057057350374175801344", "upper": "326676295606174328920832216361/1267650600228229401496703205376", "approx": 0.2577021582661335 }, "s": { "lower": "1267476888184803460528034518899/1267650600228229401496703205376", "upper": "39608652755775108141501088695/39614081257132168796771975168", "approx": 0.9998629653601753 }, "contact_left": { "lower": "2535127488413032862024737724275/1267650600228229401496703205376", "upper": "79222734012907276938273063863/39614081257132168796771975168", "approx": 1.9998629653601752 }, "contact_right": "2", "contact_diameter": { "lower": "5428501357060655270886473/39614081257132168796771975168", "upper": "173712043425940968668686477/1267650600228229401496703205376", "approx": 0.0001370346398247795 }, "contact_dimension": { "lower": "324004546623256983542760028243/633825300114114700748351602688", "upper": "648009093246513967085520531147/1267650600228229401496703205376", "approx": 0.5111890398898921 }, "canopy_dimension": { "lower": "324004546623256983542760028243/316912650057057350374175801344", "upper": "648009093246513967085520531147/633825300114114700748351602688", "approx": 1.0223780797797841 }, "radius_excess_over_2cos": { "lower": "342192826585661857813110769/1267650600228229401496703205376", "upper": "171096413292830928906869959/633825300114114700748351602688", "approx": 0.0002699425429405019 }, "normalized_radial_derivative": { "lower": "231949416628858003099328192999/158456325028528675187087900672", "upper": "463898833257716006198656899097/316912650057057350374175801344", "approx": 1.4638066141386123 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 10, "theta_over_pi": "1/20", "theta_degrees": 9.0, "R_isolating_bracket": [ "11941139124556859039363401/6044629098073145873530880", "2985284781139214759840851/1511157274518286468382720" ], "R": { "lower": "2504238379734266604811903513395/1267650600228229401496703205376", "upper": "2504238379734266604811904142541/1267650600228229401496703205376", "approx": 1.9754957551263734 }, "r": { "lower": "641687331870342163228509637367/1267650600228229401496703205376", "upper": "641687331870342163228509798581/1267650600228229401496703205376", "approx": 0.5062020494880939 }, "p": { "lower": "324823442523313901666624983507/1267650600228229401496703205376", "upper": "324823442523313901666625146721/1267650600228229401496703205376", "approx": 0.2562405149059467 }, "s": { "lower": "1267574191991879372090889811521/1267650600228229401496703205376", "upper": "1267574191991879372090890129981/1267650600228229401496703205376", "approx": 0.9999397245295065 }, "contact_left": { "lower": "2535224792220108773587593016897/1267650600228229401496703205376", "upper": "2535224792220108773587593335357/1267650600228229401496703205376", "approx": 1.9999397245295065 }, "contact_right": "2", "contact_diameter": { "lower": "76408236350029405813075395/1267650600228229401496703205376", "upper": "76408236350029405813393855/1267650600228229401496703205376", "approx": 6.0275470493425213e-05 }, "contact_dimension": { "lower": "161325540696798134847434800537/316912650057057350374175801344", "upper": "645302162787192539389739672851/1267650600228229401496703205376", "approx": 0.5090536482773814 }, "canopy_dimension": { "lower": "161325540696798134847434800537/158456325028528675187087900672", "upper": "645302162787192539389739672851/633825300114114700748351602688", "approx": 1.0181072965547628 }, "radius_excess_over_2cos": { "lower": "150944146566175758422582475/1267650600228229401496703205376", "upper": "150944146566175758423211623/1267650600228229401496703205376", "approx": 0.00011907393609800649 }, "normalized_radial_derivative": { "lower": "29117769337451077142523973471/19807040628566084398385987584", "upper": "465884309399217234280384094823/316912650057057350374175801344", "approx": 1.4700716721637297 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 11, "theta_over_pi": "1/22", "theta_degrees": 8.181818181818182, "R_isolating_bracket": [ "11966528939440380767978941/6044629098073145873530880", "186977014678755949499671/94447329657392904273920" ], "R": { "lower": "2509563009840527340832857207603/1267650600228229401496703205376", "upper": "2509563009840527340832857836749/1267650600228229401496703205376", "approx": 1.9796961476519654 }, "r": { "lower": "640325840776998358811523163497/1267650600228229401496703205376", "upper": "160081460194249589702880831007/316912650057057350374175801344", "approx": 0.5051280223917484 }, "p": { "lower": "323446525638018746814298281263/1267650600228229401496703205376", "upper": "161723262819009373407149221721/633825300114114700748351602688", "approx": 0.25515431900539864 }, "s": { "lower": "633808246991509067700156522383/633825300114114700748351602688", "upper": "1267616493983018135400313362563/1267650600228229401496703205376", "approx": 0.999973094916529 }, "contact_left": { "lower": "1267633547105623768448508125071/633825300114114700748351602688", "upper": "2535267094211247536897016567939/1267650600228229401496703205376", "approx": 1.9999730949165289 }, "contact_right": "2", "contact_diameter": { "lower": "34106245211266096389842813/1267650600228229401496703205376", "upper": "17053122605633048195080305/633825300114114700748351602688", "approx": 2.6905083471049172e-05 }, "contact_dimension": { "lower": "80411904862563892804075619577/158456325028528675187087900672", "upper": "321647619450255571216302712197/633825300114114700748351602688", "approx": 0.5074704644834243 }, "canopy_dimension": { "lower": "80411904862563892804075619577/79228162514264337593543950336", "upper": "321647619450255571216302712197/316912650057057350374175801344", "approx": 1.0149409289668485 }, "radius_excess_over_2cos": { "lower": "67520002255616782294065389/1267650600228229401496703205376", "upper": "67520002255616782294694537/1267650600228229401496703205376", "approx": 5.326389009989061e-05 }, "normalized_radial_derivative": { "lower": "467432066227359999615015792969/316912650057057350374175801344", "upper": "934864132454719999230032634371/633825300114114700748351602688", "approx": 1.4749555315737726 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 12, "theta_over_pi": "1/24", "theta_degrees": 7.5, "R_isolating_bracket": [ "1198597856123165390148475/604462909807314587353088", "11985978561231653901484753/6044629098073145873530880" ], "R": { "lower": "1198597856123165390148475/604462909807314587353088", "upper": "1256820945582204272140327636173/633825300114114700748351602688", "approx": 1.9829138176654113 }, "r": { "lower": "639286785403866451466555682979/1267650600228229401496703205376", "upper": "159821696350966612866638960747/316912650057057350374175801344", "approx": 0.5043083522295249 }, "p": { "lower": "322397665349133719978807052389/1267650600228229401496703205376", "upper": "161198832674566859989403606889/633825300114114700748351602688", "approx": 0.2543269141284585 }, "s": { "lower": "1267635196665690517855848762335/1267650600228229401496703205376", "upper": "1267635196665690517855849079617/1267650600228229401496703205376", "approx": 0.999987848731712 }, "contact_left": { "lower": "2535285796893919919352551967711/1267650600228229401496703205376", "upper": "2535285796893919919352552284993/1267650600228229401496703205376", "approx": 1.999987848731712 }, "contact_right": "2", "contact_diameter": { "lower": "15403562538883640854125759/1267650600228229401496703205376", "upper": "15403562538883640854443041/1267650600228229401496703205376", "approx": 1.2151268288052216e-05 }, "contact_dimension": { "lower": "160442283490217563234573677043/316912650057057350374175801344", "upper": "80221141745108781617286896717/158456325028528675187087900672", "approx": 0.5062665799592769 }, "canopy_dimension": { "lower": "160442283490217563234573677043/158456325028528675187087900672", "upper": "80221141745108781617286896717/79228162514264337593543950336", "approx": 1.0125331599185539 }, "radius_excess_over_2cos": { "lower": "15271968499812838355658033/633825300114114700748351602688", "upper": "15271968499812838355972607/633825300114114700748351602688", "approx": 2.4094917790538267e-05 }, "normalized_radial_derivative": { "lower": "937299969113585040858070716905/633825300114114700748351602688", "upper": "937299969113585040858071773227/633825300114114700748351602688", "approx": 1.4787986041971382 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 13, "theta_over_pi": "1/26", "theta_degrees": 6.923076923076923, "R_isolating_bracket": [ "750073774146733789833463/377789318629571617095680", "12001180386347740637335411/6044629098073145873530880" ], "R": { "lower": "2516829944958993697306922555801/1267650600228229401496703205376", "upper": "629207486239748424326730796237/316912650057057350374175801344", "approx": 1.9854287486677011 }, "r": { "lower": "319238502282912852273330157169/633825300114114700748351602688", "upper": "638477004565825704546660473943/1267650600228229401496703205376", "approx": 0.5036695477845974 }, "p": { "lower": "321581424160533731930410772247/1267650600228229401496703205376", "upper": "10049419505016679122825341657/39614081257132168796771975168", "approx": 0.2536830133655408 }, "s": { "lower": "1267643576680077827299829585565/1267650600228229401496703205376", "upper": "633821788340038913649914951225/633825300114114700748351602688", "approx": 0.9999944593974472 }, "contact_left": { "lower": "2535294176908307228796532790941/1267650600228229401496703205376", "upper": "1267647088454153614398266553913/633825300114114700748351602688", "approx": 1.9999944593974472 }, "contact_right": "2", "contact_diameter": { "lower": "3511774075787098436651463/633825300114114700748351602688", "upper": "7023548151574196873619811/1267650600228229401496703205376", "approx": 5.540602552714185e-06 }, "contact_dimension": { "lower": "640583075142436421553915548785/1267650600228229401496703205376", "upper": "320291537571218210776958006315/633825300114114700748351602688", "approx": 0.5053309445261218 }, "canopy_dimension": { "lower": "640583075142436421553915548785/633825300114114700748351602688", "upper": "320291537571218210776958006315/316912650057057350374175801344", "approx": 1.0106618890522436 }, "radius_excess_over_2cos": { "lower": "13944754417787302655622315/1267650600228229401496703205376", "upper": "1743094302223412832031433/158456325028528675187087900672", "approx": 1.1000471593100394e-05 }, "normalized_radial_derivative": { "lower": "939237548236175853891531944557/633825300114114700748351602688", "upper": "939237548236175853891533007237/633825300114114700748351602688", "approx": 1.481855565038307 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 14, "theta_over_pi": "1/28", "theta_degrees": 6.428571428571429, "R_isolating_bracket": [ "12013274068942251033933127/6044629098073145873530880", "1201327406894225103393313/604462909807314587353088" ], "R": { "lower": "1259683087011518982015746257715/633825300114114700748351602688", "upper": "1201327406894225103393313/604462909807314587353088", "approx": 1.9874294806230108 }, "r": { "lower": "318917127019483407761939355519/633825300114114700748351602688", "upper": "318917127019483407761939435161/633825300114114700748351602688", "approx": 0.5031625070221482 }, "p": { "lower": "320934282326848278419262729687/1267650600228229401496703205376", "upper": "80233570581712069604815722495/316912650057057350374175801344", "approx": 0.2531725084728133 }, "s": { "lower": "316911843063334801637817059169/316912650057057350374175801344", "upper": "1267647372253339206551268553243/1267650600228229401496703205376", "approx": 0.9999974535768061 }, "contact_left": { "lower": "633824493120392152011992860513/316912650057057350374175801344", "upper": "2535297972481568608047971758619/1267650600228229401496703205376", "approx": 1.9999974535768061 }, "contact_right": "2", "contact_diameter": { "lower": "3227974890194945434652133/1267650600228229401496703205376", "upper": "806993722548736358742175/316912650057057350374175801344", "approx": 2.5464231939098807e-06 }, "contact_dimension": { "lower": "639643710011534703717193566325/1267650600228229401496703205376", "upper": "639643710011534703717194028817/1267650600228229401496703205376", "approx": 0.5045899160986257 }, "canopy_dimension": { "lower": "639643710011534703717193566325/633825300114114700748351602688", "upper": "639643710011534703717194028817/633825300114114700748351602688", "approx": 1.0091798321972514 }, "radius_excess_over_2cos": { "lower": "1603843114871065249850129/316912650057057350374175801344", "upper": "200480389358883156250927/39614081257132168796771975168", "approx": 5.060836525718703e-06 }, "normalized_radial_derivative": { "lower": "940796451017028907824450604267/633825300114114700748351602688", "upper": "235199112754257226956112918027/158456325028528675187087900672", "approx": 1.4843150799560174 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 15, "theta_over_pi": "1/30", "theta_degrees": 6.0, "R_isolating_bracket": [ "12023046139685225722114471/6044629098073145873530880", "6011523069842612861057237/3022314549036572936765440" ], "R": { "lower": "2521415525793315049358380708659/1267650600228229401496703205376", "upper": "2521415525793315049358381337805/1267650600228229401496703205376", "approx": 1.989046134115628 }, "r": { "lower": "637315836212041262608045733247/1267650600228229401496703205376", "upper": "637315836212041262608045892271/1267650600228229401496703205376", "approx": 0.5027535474659169 }, "p": { "lower": "320412797511811049612599024387/1267650600228229401496703205376", "upper": "320412797511811049612599184287/1267650600228229401496703205376", "approx": 0.25276112948956403 }, "s": { "lower": "1267649106755630655965261025013/1267650600228229401496703205376", "upper": "1267649106755630655965261341321/1267650600228229401496703205376", "approx": 0.9999988218578538 }, "contact_left": { "lower": "2535299706983860057461964230389/1267650600228229401496703205376", "upper": "2535299706983860057461964546697/1267650600228229401496703205376", "approx": 1.999998821857854 }, "contact_right": "2", "contact_diameter": { "lower": "1493472598745531441864055/1267650600228229401496703205376", "upper": "1493472598745531442180363/1267650600228229401496703205376", "approx": 1.1781421461691768e-06 }, "contact_dimension": { "lower": "159721842948082339036066246185/316912650057057350374175801344", "upper": "638887371792329356144265446135/1267650600228229401496703205376", "approx": 0.5039932704463701 }, "canopy_dimension": { "lower": "159721842948082339036066246185/158456325028528675187087900672", "upper": "638887371792329356144265446135/633825300114114700748351602688", "approx": 1.0079865408927402 }, "radius_excess_over_2cos": { "lower": "2970585898942419739690647/1267650600228229401496703205376", "upper": "2970585898942419740319795/1267650600228229401496703205376", "approx": 2.3433790812764903e-06 }, "normalized_radial_derivative": { "lower": "235516268089170610135042077793/158456325028528675187087900672", "upper": "471032536178341220270084691637/316912650057057350374175801344", "approx": 1.486316611512781 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 16, "theta_over_pi": "1/32", "theta_degrees": 5.625, "R_isolating_bracket": [ "6015525854461587889197383/3022314549036572936765440", "12031051708923175778394769/6044629098073145873530880" ], "R": { "lower": "2523094415347165593001214030643/1267650600228229401496703205376", "upper": "2523094415347165593001214659789/1267650600228229401496703205376", "approx": 1.9903705444488446 }, "r": { "lower": "636891760563737512695601816247/1267650600228229401496703205376", "upper": "159222940140934378173900493765/316912650057057350374175801344", "approx": 0.5024190107661138 }, "p": { "lower": "159993264153760831159869606879/633825300114114700748351602688", "upper": "79996632076880415579934843335/316912650057057350374175801344", "approx": 0.2524248623792005 }, "s": { "lower": "1267649905312794760867661227157/1267650600228229401496703205376", "upper": "158456238164099345108457692907/158456325028528675187087900672", "approx": 0.9999994518083811 }, "contact_left": { "lower": "2535300505541024162364364432533/1267650600228229401496703205376", "upper": "316912563192628020295545593579/158456325028528675187087900672", "approx": 1.9999994518083812 }, "contact_right": "2", "contact_diameter": { "lower": "86864429330078630207765/158456325028528675187087900672", "upper": "694915434640629041978219/1267650600228229401496703205376", "approx": 5.481916188226594e-07 }, "contact_dimension": { "lower": "319134772632584231820082983517/633825300114114700748351602688", "upper": "638269545265168463640166427533/1267650600228229401496703205376", "approx": 0.5035058912528844 }, "canopy_dimension": { "lower": "319134772632584231820082983517/316912650057057350374175801344", "upper": "638269545265168463640166427533/633825300114114700748351602688", "approx": 1.0070117825057687 }, "radius_excess_over_2cos": { "lower": "1383139211991574304084857/1267650600228229401496703205376", "upper": "1383139211991574304714005/1267650600228229401496703205376", "approx": 1.0911044508183502e-06 }, "normalized_radial_derivative": { "lower": "235777233595777576861987090013/158456325028528675187087900672", "upper": "117888616797888788430993679461/79228162514264337593543950336", "approx": 1.4879635353989684 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 17, "theta_over_pi": "1/34", "theta_degrees": 5.294117647058823, "R_isolating_bracket": [ "12037690637553534268466239/6044629098073145873530880", "6018845318776767134233121/3022314549036572936765440" ], "R": { "lower": "631121674898166737454562751283/316912650057057350374175801344", "upper": "2524486699592666949818251634279/1267650600228229401496703205376", "approx": 1.9914688630590758 }, "r": { "lower": "636540507231935213569151906085/1267650600228229401496703205376", "upper": "636540507231935213569152064723/1267650600228229401496703205376", "approx": 0.5021419207448264 }, "p": { "lower": "319633672933330023316321955721/1267650600228229401496703205376", "upper": "319633672933330023316322115039/1267650600228229401496703205376", "approx": 0.25214650856930354 }, "s": { "lower": "633825137647014474971483008779/633825300114114700748351602688", "upper": "633825137647014474971483166741/633825300114114700748351602688", "approx": 0.9999997436721125 }, "contact_left": { "lower": "1267650437761129175719834611467/633825300114114700748351602688", "upper": "1267650437761129175719834769429/633825300114114700748351602688", "approx": 1.9999997436721124 }, "contact_right": "2", "contact_diameter": { "lower": "162467100225776868435947/633825300114114700748351602688", "upper": "162467100225776868593909/633825300114114700748351602688", "approx": 2.563278875054784e-07 }, "contact_dimension": { "lower": "637758405239115649474954108227/1267650600228229401496703205376", "upper": "637758405239115649474954567989/1267650600228229401496703205376", "approx": 0.5031026728692376 }, "canopy_dimension": { "lower": "637758405239115649474954108227/633825300114114700748351602688", "upper": "637758405239115649474954567989/633825300114114700748351602688", "approx": 1.0062053457384752 }, "radius_excess_over_2cos": { "lower": "161774085685566384198321/316912650057057350374175801344", "upper": "647096342742265537422433/1267650600228229401496703205376", "approx": 5.104690067008697e-07 }, "normalized_radial_derivative": { "lower": "117997112134802620550581118397/79228162514264337593543950336", "upper": "943976897078420964404650025771/633825300114114700748351602688", "approx": 1.4893329390740102 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 18, "theta_over_pi": "1/36", "theta_degrees": 5.0, "R_isolating_bracket": [ "6021628184248560560674769/3022314549036572936765440", "12043256368497121121349541/6044629098073145873530880" ], "R": { "lower": "2525653917970647455388042631577/1267650600228229401496703205376", "upper": "631413479492661863847010815181/316912650057057350374175801344", "approx": 1.9923896360053202 }, "r": { "lower": "636246333207107959920270288311/1267650600228229401496703205376", "upper": "636246333207107959920270446803/1267650600228229401496703205376", "approx": 0.5019098583573087 }, "p": { "lower": "319338306980336583625304082797/1267650600228229401496703205376", "upper": "319338306980336583625304241895/1267650600228229401496703205376", "approx": 0.25191350591625367 }, "s": { "lower": "1267650447642470088194008873477/1267650600228229401496703205376", "upper": "158456305955308761024251148657/158456325028528675187087900672", "approx": 0.9999998796310597 }, "contact_left": { "lower": "2535301047870699489690712078853/1267650600228229401496703205376", "upper": "316912630983837436211339049329/158456325028528675187087900672", "approx": 1.9999998796310599 }, "contact_right": "2", "contact_diameter": { "lower": "19073219914162836752015/158456325028528675187087900672", "upper": "152585759313302694331899/1267650600228229401496703205376", "approx": 1.2036894021572739e-07 }, "contact_dimension": { "lower": "318665369644365187355470716201/633825300114114700748351602688", "upper": "79666342411091296838867736443/158456325028528675187087900672", "approx": 0.5027653039204845 }, "canopy_dimension": { "lower": "318665369644365187355470716201/316912650057057350374175801344", "upper": "79666342411091296838867736443/79228162514264337593543950336", "approx": 1.005530607840969 }, "radius_excess_over_2cos": { "lower": "304010285457826527952969/1267650600228229401496703205376", "upper": "152005142728913264291059/633825300114114700748351602688", "approx": 2.3982182898275923e-07 }, "normalized_radial_derivative": { "lower": "944705691828071031853596241039/633825300114114700748351602688", "upper": "944705691828071031853597322135/633825300114114700748351602688", "approx": 1.4904827744458686 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 19, "theta_over_pi": "1/38", "theta_degrees": 4.7368421052631575, "R_isolating_bracket": [ "3011991983448779819899051/1511157274518286468382720", "12047967933795119279596207/6044629098073145873530880" ], "R": { "lower": "631660501207357549686093460275/316912650057057350374175801344", "upper": "2526642004829430198744374470247/1267650600228229401496703205376", "approx": 1.9931690990991764 }, "r": { "lower": "635997518123851600340752886223/1267650600228229401496703205376", "upper": "635997518123851600340753044591/1267650600228229401496703205376", "approx": 0.5017135778654984 }, "p": { "lower": "319088590331494772182215690805/1267650600228229401496703205376", "upper": "319088590331494772182215849717/1267650600228229401496703205376", "approx": 0.25171651421459956 }, "s": { "lower": "1267650528305917932528389069549/1267650600228229401496703205376", "upper": "1267650528305917932528389385205/1267650600228229401496703205376", "approx": 0.9999999432633002 }, "contact_left": { "lower": "2535301128534147334025092274925/1267650600228229401496703205376", "upper": "2535301128534147334025092590581/1267650600228229401496703205376", "approx": 1.9999999432633002 }, "contact_right": "2", "contact_diameter": { "lower": "71922311468968313820171/1267650600228229401496703205376", "upper": "71922311468968314135827/1267650600228229401496703205376", "approx": 5.6736699730997904e-08 }, "contact_dimension": { "lower": "318484654441414266207644109057/633825300114114700748351602688", "upper": "636969308882828532415288676743/1267650600228229401496703205376", "approx": 0.5024801856033104 }, "canopy_dimension": { "lower": "318484654441414266207644109057/316912650057057350374175801344", "upper": "636969308882828532415288676743/633825300114114700748351602688", "approx": 1.0049603712066209 }, "radius_excess_over_2cos": { "lower": "71676664377866967804253/633825300114114700748351602688", "upper": "143353328755733936237655/1267650600228229401496703205376", "approx": 1.1308583668869357e-07 }, "normalized_radial_derivative": { "lower": "945323192685852320481829450299/633825300114114700748351602688", "upper": "59082699542865770030114408349/39614081257132168796771975168", "approx": 1.4914570190171568 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 20, "theta_over_pi": "1/40", "theta_degrees": 4.5, "R_isolating_bracket": [ "12051991371097037489435359/6044629098073145873530880", "6025995685548518744717681/3022314549036572936765440" ], "R": { "lower": "631871445196972359126108549939/316912650057057350374175801344", "upper": "2527485780787889436504434828903/1267650600228229401496703205376", "approx": 1.9938347209655702 }, "r": { "lower": "158946299171469704093376122341/316912650057057350374175801344", "upper": "317892598342939408186752323813/633825300114114700748351602688", "approx": 0.5015460857837414 }, "p": { "lower": "79718894199262168500270586027/316912650057057350374175801344", "upper": "79718894199262168500270625715/316912650057057350374175801344", "approx": 0.2515484761649921 }, "s": { "lower": "633825283107078662923092477405/633825300114114700748351602688", "upper": "158456320776769665730773158795/158456325028528675187087900672", "approx": 0.9999999731676283 }, "contact_left": { "lower": "1267650583221193363671444080093/633825300114114700748351602688", "upper": "316912645805298340917861059467/158456325028528675187087900672", "approx": 1.9999999731676283 }, "contact_right": "2", "contact_diameter": { "lower": "4251759009456314741877/158456325028528675187087900672", "upper": "17007036037825259125283/633825300114114700748351602688", "approx": 2.6832371687850407e-08 }, "contact_dimension": { "lower": "318330550862578395768168860305/633825300114114700748351602688", "upper": "636661101725156791536338178797/1267650600228229401496703205376", "approx": 0.5022370530259138 }, "canopy_dimension": { "lower": "318330550862578395768168860305/316912650057057350374175801344", "upper": "636661101725156791536338178797/633825300114114700748351602688", "approx": 1.0044741060518276 }, "radius_excess_over_2cos": { "lower": "16954609476464361496299/316912650057057350374175801344", "upper": "67818437905857446614345/1267650600228229401496703205376", "approx": 5.3499314317089685e-08 }, "normalized_radial_derivative": { "lower": "472925383282408981783531873629/316912650057057350374175801344", "upper": "472925383282408981783532416207/316912650057057350374175801344", "approx": 1.4922893838326206 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 21, "theta_over_pi": "1/42", "theta_degrees": 4.285714285714286, "R_isolating_bracket": [ "12055454331736831979723209/6044629098073145873530880", "3013863582934207994930803/1511157274518286468382720" ], "R": { "lower": "632053004067764016498512180019/316912650057057350374175801344", "upper": "2528212016271056065994049349223/1267650600228229401496703205376", "approx": 1.9944076197462248 }, "r": { "lower": "158900641433260471512077450845/316912650057057350374175801344", "upper": "635602565733041886048309961551/1267650600228229401496703205376", "approx": 0.5014020153649651 }, "p": { "lower": "318692407429689895762772203761/1267650600228229401496703205376", "upper": "318692407429689895762772362377/1267650600228229401496703205376", "approx": 0.2514039810120487 }, "s": { "lower": "1267650584094180048398275364127/1267650600228229401496703205376", "upper": "1267650584094180048398275679587/1267650600228229401496703205376", "approx": 0.9999999872724792 }, "contact_left": { "lower": "2535301184322409449894978569503/1267650600228229401496703205376", "upper": "2535301184322409449894978884963/1267650600228229401496703205376", "approx": 1.9999999872724792 }, "contact_right": "2", "contact_diameter": { "lower": "16134049353098427525789/1267650600228229401496703205376", "upper": "16134049353098427841249/1267650600228229401496703205376", "approx": 1.2727520777565706e-08 }, "contact_dimension": { "lower": "636396149735700140424755348451/1267650600228229401496703205376", "upper": "318198074867850070212377903125/633825300114114700748351602688", "approx": 0.502028042759671 }, "canopy_dimension": { "lower": "636396149735700140424755348451/633825300114114700748351602688", "upper": "318198074867850070212377903125/316912650057057350374175801344", "approx": 1.004056085519342 }, "radius_excess_over_2cos": { "lower": "16088935483590575753025/633825300114114700748351602688", "upper": "32177870967181152135199/1267650600228229401496703205376", "approx": 2.538386441925544e-08 }, "normalized_radial_derivative": { "lower": "118288120308229038723225789707/79228162514264337593543950336", "upper": "946304962465832309785807404425/633825300114114700748351602688", "approx": 1.4930059786118641 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 22, "theta_over_pi": "1/44", "theta_degrees": 4.090909090909091, "R_isolating_bracket": [ "3014614056201995515789279/1511157274518286468382720", "12058456224807982063157119/6044629098073145873530880" ], "R": { "lower": "2528841558876850919971407213363/1267650600228229401496703205376", "upper": "2528841558876850919971407842509/1267650600228229401496703205376", "approx": 1.9949042412961073 }, "r": { "lower": "635444335616142355908850350059/1267650600228229401496703205376", "upper": "635444335616142355908850508151/1267650600228229401496703205376", "approx": 0.5012771938117145 }, "p": { "lower": "318533753381209141320648640923/1267650600228229401496703205376", "upper": "79633438345302285330162199855/316912650057057350374175801344", "approx": 0.25127882503574717 }, "s": { "lower": "1267650592554857683149183206745/1267650600228229401496703205376", "upper": "1267650592554857683149183522125/1267650600228229401496703205376", "approx": 0.9999999939467771 }, "contact_left": { "lower": "2535301192783087084645886412121/1267650600228229401496703205376", "upper": "2535301192783087084645886727501/1267650600228229401496703205376", "approx": 1.999999993946777 }, "contact_right": "2", "contact_diameter": { "lower": "7673371718347519683251/1267650600228229401496703205376", "upper": "7673371718347519998631/1267650600228229401496703205376", "approx": 6.0532229598329354e-09 }, "contact_dimension": { "lower": "636166714020106595761740401317/1267650600228229401496703205376", "upper": "636166714020106595761740858789/1267650600228229401496703205376", "approx": 0.5018470498933778 }, "canopy_dimension": { "lower": "636166714020106595761740401317/633825300114114700748351602688", "upper": "636166714020106595761740858789/633825300114114700748351602688", "approx": 1.0036940997867556 }, "radius_excess_over_2cos": { "lower": "15307641785973065439807/1267650600228229401496703205376", "upper": "15307641785973066068955/1267650600228229401496703205376", "approx": 1.2075600156081699e-08 }, "normalized_radial_derivative": { "lower": "473349365102525434315224645849/316912650057057350374175801344", "upper": "473349365102525434315225190003/316912650057057350374175801344", "approx": 1.4936272345622776 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 23, "theta_over_pi": "1/46", "theta_degrees": 3.9130434782608696, "R_isolating_bracket": [ "12061075379782482007282687/6044629098073145873530880", "1206107537978248200728269/604462909807314587353088" ], "R": { "lower": "1264695417743079585326845080371/633825300114114700748351602688", "upper": "1206107537978248200728269/604462909807314587353088", "approx": 1.9953375441393428 }, "r": { "lower": "158826586001895024482071448281/316912650057057350374175801344", "upper": "635306344007580097928285951147/1267650600228229401496703205376", "approx": 0.5011683376264712 }, "p": { "lower": "79598856077457484632299023801/316912650057057350374175801344", "upper": "159197712154914969264598126799/633825300114114700748351602688", "approx": 0.25116970263928057 }, "s": { "lower": "316912649142491862264344957259/316912650057057350374175801344", "upper": "1267650596569967449057380144347/1267650600228229401496703205376", "approx": 0.9999999971141401 }, "contact_left": { "lower": "633825299199549212638520758603/316912650057057350374175801344", "upper": "2535301196798196850554083349723/1267650600228229401496703205376", "approx": 1.9999999971141402 }, "contact_right": "2", "contact_diameter": { "lower": "3658261952439323061029/1267650600228229401496703205376", "upper": "914565488109830844085/316912650057057350374175801344", "approx": 2.8858598353368705e-09 }, "contact_dimension": { "lower": "635966712812073564900250558017/1267650600228229401496703205376", "upper": "635966712812073564900251015207/1267650600228229401496703205376", "approx": 0.5016892767593636 }, "canopy_dimension": { "lower": "635966712812073564900250558017/633825300114114700748351602688", "upper": "635966712812073564900251015207/633825300114114700748351602688", "approx": 1.0033785535187272 }, "radius_excess_over_2cos": { "lower": "3649733709999338209005/633825300114114700748351602688", "upper": "3649733709999338523579/633825300114114700748351602688", "approx": 5.758264476571439e-09 }, "normalized_radial_derivative": { "lower": "473521152508629909914698429179/316912650057057350374175801344", "upper": "947042305017259819829397948077/633825300114114700748351602688", "approx": 1.4941693000370182 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true }, { "m": 24, "theta_over_pi": "1/48", "theta_degrees": 3.75, "R_isolating_bracket": [ "12063374182993421984032747/6044629098073145873530880", "1206337418299342198403275/604462909807314587353088" ], "R": { "lower": "1264936464730651045032912171827/633825300114114700748351602688", "upper": "1206337418299342198403275/604462909807314587353088", "approx": 1.9957178492289758 }, "r": { "lower": "635185279681680794192077444657/1267650600228229401496703205376", "upper": "158796319920420198548019400655/316912650057057350374175801344", "approx": 0.5010728347127523 }, "p": { "lower": "79568522164478040635577607293/316912650057057350374175801344", "upper": "318274088657912162542310587475/1267650600228229401496703205376", "approx": 0.2510739856870732 }, "s": { "lower": "1267650598480346430565483444637/1267650600228229401496703205376", "upper": "1267650598480346430565483759889/1267650600228229401496703205376", "approx": 0.9999999986211635 }, "contact_left": { "lower": "2535301198708575832062186650013/1267650600228229401496703205376", "upper": "2535301198708575832062186965265/1267650600228229401496703205376", "approx": 1.9999999986211634 }, "contact_right": "2", "contact_diameter": { "lower": "1747882970931219445487/1267650600228229401496703205376", "upper": "1747882970931219760739/1267650600228229401496703205376", "approx": 1.3788365426691934e-09 }, "contact_dimension": { "lower": "9934239270684201326545242233/19807040628566084398385987584", "upper": "635791313323788884898895959839/1267650600228229401496703205376", "approx": 0.50155091096026 }, "canopy_dimension": { "lower": "9934239270684201326545242233/9903520314283042199192993792", "upper": "635791313323788884898895959839/633825300114114700748351602688", "approx": 1.00310182192052 }, "radius_excess_over_2cos": { "lower": "1744140621725402773973/633825300114114700748351602688", "upper": "1744140621725403088547/633825300114114700748351602688", "approx": 2.7517686993740794e-09 }, "normalized_radial_derivative": { "lower": "59208990907817962214483170327/39614081257132168796771975168", "upper": "947343854525087395431731816247/633825300114114700748351602688", "approx": 1.494645061272446 }, "proof": "analytic all-m theorem; this record certifies the numerical presentation", "passed": true } ] }