{ "_public_export": { "status": "RIEMANN HYPOTHESIS OPEN", "source_path": "research/full_ground_capture/priority2/experiments/tail_certificates.json", "source_sha256": "afd934e9e37d121c92920410e4ee763ac373284b0bee1b735f84461bfa43dcc6", "public_credit": "@ykbballer91", "license": "CC-BY-4.0", "historical_snapshot": true, "scope": "Priority 2 only; no growing-m, full-ground or RH claim" }, "method": "Arb interval enclosure of L*tau < 1", "python_flint_version": "0.9.0", "RH_assumed": false, "noncertificate_does_not_mean_rank_loss": true, "records": [ { "a": "0.5", "N": 2, "m": 2, "precision_bits": 256, "status": "TAIL_MAJORANT_PRECONDITION_NOT_VERIFIED", "Y": "[8.539734222673567065463550869546574495034888535765114961879601130179228611157 +/- 4.60e-76]" }, { "a": "1", "N": 2, "m": 2, "precision_bits": 256, "status": "SUFFICIENT_CRITERION_INCONCLUSIVE", "inverse_norm_upper_enclosure": "[12.92739589317147233572267945700662169159226293364963235272297435379777932574 +/- 6.39e-75]", "ideal_sigma_min_lower_enclosure": "[0.0773550998409680823062736640455656576825651449791987410896617725510769114752 +/- 2.51e-77]", "tail_norm_upper_enclosure": "[0.087247567092079482559493298175408037572202734849232616000700493981913254769 +/- 1.88e-76]", "eta_upper_enclosure": "[1.12788384051535079978455884428511514670346638707485097109947992322323490642 +/- 3.79e-75]", "actual_sigma_min_lower_enclosure": "[-0.009892467251111400253219634129842379889637589870033874911038721430836343294 +/- 2.50e-76]", "Xi_nonzero_rows_certified": 3 }, { "a": "1", "N": 4, "m": 4, "precision_bits": 256, "status": "SUFFICIENT_CRITERION_INCONCLUSIVE", "inverse_norm_upper_enclosure": "[13.1995893464532845225382494928831657956008293031359564364480998395680849807 +/- 1.24e-74]", "ideal_sigma_min_lower_enclosure": "[0.0757599326579579481940049437521926259199196088952344232078415440340966721476 +/- 5.75e-77]", "tail_norm_upper_enclosure": "[1537789.10275542161790485933339266650321379672272113569901315722206141834044 +/- 2.97e-69]", "eta_upper_enclosure": "[20298184.6578224184306420784743783378372504254749000965989735530363230801911 +/- 5.39e-68]", "actual_sigma_min_lower_enclosure": "[-1537789.02699548895994691113938772275102117080280152680377873401421987430634 +/- 5.14e-69]", "Xi_nonzero_rows_certified": 5 }, { "a": "1.5", "N": 4, "m": 4, "precision_bits": 256, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[19.3339332541968274889903808947049650217533671629382178907268308794474748674 +/- 6.75e-74]", "ideal_sigma_min_lower_enclosure": "[0.0517225329606912474403189501438587696312265230557955688945503260647410520022 +/- 6.21e-77]", "tail_norm_upper_enclosure": "[2.3415245726969903919663562825422421591851967798599733742085679119089685808e-8 +/- 1.43e-82]", "eta_upper_enclosure": "[4.5270879801585359407010053385547454885629673280367492645743119541689119079e-7 +/- 6.57e-81]", "actual_sigma_min_lower_enclosure": "[0.0517225095454455204704150304802959442088049312038277702948165839790619329125 +/- 4.87e-77]", "Xi_nonzero_rows_certified": 5 }, { "a": "2", "N": 4, "m": 4, "precision_bits": 256, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[28.2904523882395846165175841559807021173724042953506706280320206245263548248 +/- 2.36e-74]", "ideal_sigma_min_lower_enclosure": "[0.0353476143214911129252873821537512993418751384060485521569381563752765540018 +/- 3.51e-77]", "tail_norm_upper_enclosure": "[1.1283978582650194386319890643794012506996888733957973755592823384910251544e-51 +/- 5.46e-125]", "eta_upper_enclosure": "[3.1922885884238051482664462441079830537214641021826956950547129333900978452e-50 +/- 6.70e-124]", "actual_sigma_min_lower_enclosure": "[0.0353476143214911129252873821537512993418751384060474237590798913558379220128 +/- 6.95e-77]", "Xi_nonzero_rows_certified": 5 }, { "a": "2", "N": 6, "m": 6, "precision_bits": 256, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[29.3503655469953103370332329461133064084025059698165595140587349127011274274 +/- 4.55e-74]", "ideal_sigma_min_lower_enclosure": "[0.0340711259080850923106019877527127940035074210419593884592058602344578529288 +/- 4.32e-77]", "tail_norm_upper_enclosure": "[2.4296981175613136755551655985506453574883942853856813393405830370793416783e-41 +/- 6.30e-115]", "eta_upper_enclosure": "[7.131252791927094209779191756521736672674845877884306218101815834346908393e-40 +/- 1.09e-113]", "actual_sigma_min_lower_enclosure": "[0.0340711259080850923106019877527127940034831240607837753224503085784723464752 +/- 3.79e-77]", "Xi_nonzero_rows_certified": 7 }, { "a": "3", "N": 6, "m": 6, "precision_bits": 256, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[67.186101140328342594612228050474302069899450253341633938057828759382509080 +/- 1.55e-73]", "ideal_sigma_min_lower_enclosure": "[0.0148840308192813365371972051833502427056441362955508092000011562829365390133 +/- 4.96e-77]", "tail_norm_upper_enclosure": "[4.721738540495593972189552374002900446870139266663097426826677796653027278e-506 +/- 3.41e-579]", "eta_upper_enclosure": "[3.172352031399233102233684912029824514574137770253252131942533166587780285e-504 +/- 5.86e-577]", "actual_sigma_min_lower_enclosure": "[0.0148840308192813365371972051833502427056441362955508092000011562829365390133 +/- 4.96e-77]", "Xi_nonzero_rows_certified": 7 }, { "a": "1.5 +/- 0.000001", "N": 4, "m": 4, "precision_bits": 256, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[19.334 +/- 3.81e-4]", "ideal_sigma_min_lower_enclosure": "[0.05172 +/- 3.38e-6]", "tail_norm_upper_enclosure": "[2.342e-8 +/- 8.14e-12]", "eta_upper_enclosure": "[4.527e-7 +/- 8.14e-11]", "actual_sigma_min_lower_enclosure": "[0.05172 +/- 3.35e-6]", "Xi_nonzero_rows_certified": 5 }, { "a": "2 +/- 0.000001", "N": 6, "m": 6, "precision_bits": 256, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[29.35 +/- 9.77e-4]", "ideal_sigma_min_lower_enclosure": "[0.034071 +/- 8.35e-7]", "tail_norm_upper_enclosure": "[2.43e-41 +/- 1.21e-44]", "eta_upper_enclosure": "[7.13e-40 +/- 4.05e-43]", "actual_sigma_min_lower_enclosure": "[0.034071 +/- 8.35e-7]", "Xi_nonzero_rows_certified": 7 }, { "a": "0.5", "N": 2, "m": 2, "precision_bits": 384, "status": "TAIL_MAJORANT_PRECONDITION_NOT_VERIFIED", "Y": "[8.539734222673567065463550869546574495034888535765114961879601130179228611157330807572563869710473943913774942511677 +/- 9.26e-115]" }, { "a": "1", "N": 2, "m": 2, "precision_bits": 384, "status": "SUFFICIENT_CRITERION_INCONCLUSIVE", "inverse_norm_upper_enclosure": "[12.9273958931714723357226794570066216915922629336496323527229743537977793257430874364833868528286568930237968823643 +/- 5.99e-113]", "ideal_sigma_min_lower_enclosure": "[0.077355099840968082306273664045565657682565144979198741089661772551076911475195801489038235998693443065699063471187 +/- 2.34e-115]", "tail_norm_upper_enclosure": "[0.087247567092079482559493298175408037572202734849232616000700493981913254769093589885392620053752763946206902909683 +/- 4.31e-115]", "eta_upper_enclosure": "[1.12788384051535079978455884428511514670346638707485097109947992322323490642226057304940783622377903856370322901982 +/- 5.86e-114]", "actual_sigma_min_lower_enclosure": "[-0.009892467251111400253219634129842379889637589870033874911038721430836343293897788396354384055059320880507839438496 +/- 4.45e-115]", "Xi_nonzero_rows_certified": 3 }, { "a": "1", "N": 4, "m": 4, "precision_bits": 384, "status": "SUFFICIENT_CRITERION_INCONCLUSIVE", "inverse_norm_upper_enclosure": "[13.1995893464532845225382494928831657956008293031359564364480998395680849807060732061971779088718506527967950873821 +/- 6.03e-113]", "ideal_sigma_min_lower_enclosure": "[0.075759932657957948194004943752192625919919608895234423207841544034096672147580084587188725171352471486039511224289 +/- 5.32e-115]", "tail_norm_upper_enclosure": "[1537789.10275542161790485933339266650321379672272113569901315722206141834043971690131666107838547942384444616519281 +/- 9.55e-108]", "eta_upper_enclosure": "[20298184.657822418430642078474378337837250425474900096598973553036323080191093254954733616045244658703461741230120 +/- 3.57e-106]", "actual_sigma_min_lower_enclosure": "[-1537789.02699548895994691113938772275102117080280152680377873401421987430634304475373657649119675425249197467915330 +/- 8.34e-108]", "Xi_nonzero_rows_certified": 5 }, { "a": "1.5", "N": 4, "m": 4, "precision_bits": 384, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[19.3339332541968274889903808947049650217533671629382178907268308794474748674503491138833323200674209579157864634659 +/- 7.85e-113]", "ideal_sigma_min_lower_enclosure": "[0.051722532960691247440318950143858769631226523055795568894550326064741052002184408033486221184117848544909615455722 +/- 4.65e-115]", "tail_norm_upper_enclosure": "[2.3415245726969903919663562825422421591851967798599733742085679119089685808075905386220123375656031011876699152954e-8 +/- 6.36e-121]", "eta_upper_enclosure": "[4.527087980158535940701005338554745488562967328036749264574311954168911907948741760215301789642599995696449937114e-7 +/- 5.16e-119]", "actual_sigma_min_lower_enclosure": "[0.051722509545445520470415030480295944208804931203827770294816583979061932912498599957580834963994472888878603579023 +/- 6.20e-115]", "Xi_nonzero_rows_certified": 5 }, { "a": "2", "N": 4, "m": 4, "precision_bits": 384, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[28.2904523882395846165175841559807021173724042953506706280320206245263548247928790093954853096527807210579354278325 +/- 8.11e-113]", "ideal_sigma_min_lower_enclosure": "[0.035347614321491112925287382153751299341875138406048552156938156375276554001814554250128320259799788444626503811633 +/- 3.98e-115]", "tail_norm_upper_enclosure": "[1.1283978582650194386319890643794012506996888733957973755592823384910251543513899274537709731528652335261682658600e-51 +/- 4.39e-164]", "eta_upper_enclosure": "[3.1922885884238051482664462441079830537214641021826956950547129333900978451516837494702265117877601400546728445160e-50 +/- 8.03e-163]", "actual_sigma_min_lower_enclosure": "[0.0353476143214911129252873821537512993418751384060474237590798913558379220127501748488776205709263926472509445292942 +/- 9.12e-116]", "Xi_nonzero_rows_certified": 5 }, { "a": "2", "N": 6, "m": 6, "precision_bits": 384, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[29.350365546995310337033232946113306408402505969816559514058734912701127427379716215526142388316083455956826581592 +/- 4.43e-112]", "ideal_sigma_min_lower_enclosure": "[0.0340711259080850923106019877527127940035074210419593884592058602344578529287831299984190523958432961247563703637796 +/- 9.29e-116]", "tail_norm_upper_enclosure": "[2.4296981175613136755551655985506453574883942853856813393405830370793416783423778932530839521191326189727489066338e-41 +/- 8.77e-154]", "eta_upper_enclosure": "[7.131252791927094209779191756521736672674845877884306218101815834346908393042864126464944861069484729040949231419e-40 +/- 2.77e-152]", "actual_sigma_min_lower_enclosure": "[0.034071125908085092310601987752712794003483124060783775322450308578472346475208246055565195582449890294385576946996 +/- 2.55e-115]", "Xi_nonzero_rows_certified": 7 }, { "a": "3", "N": 6, "m": 6, "precision_bits": 384, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[67.186101140328342594612228050474302069899450253341633938057828759382509079963879594389372981569571062382096144167 +/- 6.36e-112]", "ideal_sigma_min_lower_enclosure": "[0.0148840308192813365371972051833502427056441362955508092000011562829365390133232982461533655531927825229062001362788 +/- 8.90e-116]", "tail_norm_upper_enclosure": "[4.721738540495593972189552374002900446870139266663097426826677796653027278043587315693046258566033553442691373889e-506 +/- 9.02e-618]", "eta_upper_enclosure": "[3.172352031399233102233684912029824514574137770253252131942533166587780285380467658201867495036528451150909114766e-504 +/- 8.22e-616]", "actual_sigma_min_lower_enclosure": "[0.0148840308192813365371972051833502427056441362955508092000011562829365390133232982461533655531927825229062001362788 +/- 8.98e-116]", "Xi_nonzero_rows_certified": 7 }, { "a": "1.5 +/- 0.000001", "N": 4, "m": 4, "precision_bits": 384, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[19.334 +/- 4.36e-4]", "ideal_sigma_min_lower_enclosure": "[0.05172 +/- 3.53e-6]", "tail_norm_upper_enclosure": "[2.342e-8 +/- 8.14e-12]", "eta_upper_enclosure": "[4.527e-7 +/- 8.27e-11]", "actual_sigma_min_lower_enclosure": "[0.05172 +/- 3.50e-6]", "Xi_nonzero_rows_certified": 5 }, { "a": "2 +/- 0.000001", "N": 6, "m": 6, "precision_bits": 384, "status": "CERTIFIED_FULL_RANK", "inverse_norm_upper_enclosure": "[29.35 +/- 1.07e-3]", "ideal_sigma_min_lower_enclosure": "[0.03407 +/- 1.94e-6]", "tail_norm_upper_enclosure": "[2.43e-41 +/- 1.21e-44]", "eta_upper_enclosure": "[7.13e-40 +/- 4.07e-43]", "actual_sigma_min_lower_enclosure": "[0.03407 +/- 1.94e-6]", "Xi_nonzero_rows_certified": 7 } ] }