Recorded data and provenance for the associated calculation.
| experiment_key | ground-current-repair-harmonic-intersection |
|---|---|
| created_utc | 2026-10-12T04:27:38.625313+00:00 |
View complete data
{
"experiment_key": "ground-current-repair-harmonic-intersection",
"created_utc": "2026-10-12T04:27:38.625313+00:00",
"sources": {
"experiments/ground-harmonic-repair.py": "d76a036de187a8328a6c05bb7366e91737455177458f37eb094737e46e5bf9de",
"experiments/inverse-current-ground-proof.py": "8f941c0913b297522f5578c78c6361834e0179d20c702c46ccdd3c58f7ca5c2c",
"experiments/inverse-current-radial.py": "52d9f1c981635192e06d96b11f0b1e4e5e2d1bf3d5ed420b59411fd45ef7be80"
},
"cases": [
{
"depth": 2,
"c_exact": "3/10",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "z**3 - 377169*z**2/62500 + 1658961*z/1250000 - 52057161/625000000",
"discriminant": "-31494506341788888081003/4768371582031250000000",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 3,
"c_exact": "3/10",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "18100103*kappa/322265625 + z**3 + z**2*(437304*kappa/1890625 - 345439/123750) + z*(10015517/12375000 - 305383561*kappa/567187500) - 233758/3515625",
"discriminant": "(18970220085333072146922654876672*kappa**4 + 270343936154361113514776093079600*kappa**3 + 1553716532649956017383405936917500*kappa**2 - 1766285228358407143413155687250000*kappa - 323405751050341929055386192578125)/1490284197320938110351562500000000",
"remaining_necessary_blend_set": "((kappa <= 9183350/7757187) & (275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "193648*kappa/48828125 + z**4 + z**3*(437304*kappa/1890625 - 444439/123750) + z**2*(382346/171875 - 693994*kappa/1171875) + z*(605601668*kappa/3544921875 - 5474366/12890625) + 30392/5859375",
"discriminant": "4*(21205288932218091326233795873938765346270596096*kappa**6 + 120330274645513442357176278246262526915970739200*kappa**5 + 1448441782872891807905296320280250091697676287500*kappa**4 - 10926622499872657320253652185818391235576829062500*kappa**3 + 7551735086608403012169851516510291539575708984375*kappa**2 + 15434333371535422572578460357885693755018730468750*kappa - 13857497780108076417146030939529835079106689453125)/219464218314866752024272500420920550823211669921875",
"remaining_necessary_blend_set": "((kappa <= 9183350/7757187) & (275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 1) <= kappa)) | ((-94975/72618 <= kappa) & (kappa <= 275*CRootOf(21205288932218091326233795873938765346270596096*x**6 + 437564635074594335844277375440954643330802688*x**5 + 19152949195013445393789042251639670634018860*x**4 - 525397325313185626218056904502161562041260*x**3 + 1320431788929547961966724942440157526215*x**2 + 9813510858332002107605878514554365018*x - 32039715768930723246125954865970861, 0)))"
},
{
"scalar_harmonic_degree": 3,
"characteristic_polynomial": "326781*kappa/9765625 + z**4 + z**3*(437304*kappa/1890625 - 583039/123750) + z**2*(12405373/2475000 - 75721289*kappa/113437500) + z*(1100380687*kappa/3544921875 - 67509707/38671875) + 71991/781250",
"discriminant": "89*(9964567346077008054687898426838589647031081222144*kappa**6 - 37777923827263523970942415558100000337295715731200*kappa**5 + 848073954344510964123231819320789836900773692400000*kappa**4 - 10947330573528968003817266402281202810245835872500000*kappa**3 - 16688514904804803017421000726818378942558935679687500*kappa**2 + 83999563005108798676621623460048749156159191914062500*kappa - 58261275019065972589452029703744875474539311279296875)/674194078663270662218565121293067932128906250000000000",
"remaining_necessary_blend_set": "(kappa <= 9183350/7757187) & (275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 1) <= kappa)"
},
{
"scalar_harmonic_degree": 4,
"characteristic_polynomial": "36599472*kappa/244140625 + z**4 + z**3*(437304*kappa/1890625 - 761239/123750) + z**2*(15436072/1546875 - 108380902*kappa/141796875) + z*(321823292*kappa/708984375 - 43492582/7734375) + 6753096/9765625",
"discriminant": "28*(66166161550673737410361702595614877091432225792*kappa**6 - 946219995174720007466628124239475630174698681600*kappa**5 + 3293795765028826293749991671572507137671684137500*kappa**4 - 78015100455343851345471545020761987259479072187500*kappa**3 - 1320094024221372783034587142454927501644319041015625*kappa**2 + 3820730287803669979298030054837985803495904648437500*kappa - 2474582904153747612935096245439872424036008789062500)/658392654944600256072817501262761652469635009765625",
"remaining_necessary_blend_set": "(kappa <= 9183350/7757187) & (275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 1) <= kappa)"
},
{
"scalar_harmonic_degree": 5,
"characteristic_polynomial": "4744376*kappa/9765625 + z**4 + z**3*(437304*kappa/1890625 - 979039/123750) + z**2*(4994321/275000 - 100128917*kappa/113437500) + z*(2036252477*kappa/3544921875 - 196097174/12890625) + 3919804/1171875",
"discriminant": "49*(3169131954784137083156957049212007158558921444352*kappa**6 - 12427811022673457659291668400292658864119880159600*kappa**5 - 3299148367310275939447159937461819922275798770487500*kappa**4 + 41410625630028551054373920851537135317661906895000000*kappa**3 - 669132122847050676898747265655983451481530528943359375*kappa**2 + 1635431326210883773216895000501652937979500880039062500*kappa - 1025659864681034913414846891933237441104345117675781250)/56182839888605888518213760107755661010742187500000000",
"remaining_necessary_blend_set": "(kappa <= 9183350/7757187) & (275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 1) <= kappa)"
},
{
"scalar_harmonic_degree": 6,
"characteristic_polynomial": "62741952*kappa/48828125 + z**4 + z**3*(437304*kappa/1890625 - 1236439/123750) + z**2*(47684509/1546875 - 144992344*kappa/141796875) + z*(2264997448*kappa/3544921875 - 1394457428/38671875) + 24018336/1953125",
"discriminant": "-28*(251066249587713269315404282168684149316749950976*kappa**6 - 24640821529019272025503976662049480792757883084800*kappa**5 + 791578382825315686469497160169661053056727069600000*kappa**4 - 10687471262214380422631138983286174347868874240000000*kappa**3 + 93369535807708085581854755365407410925275326226562500*kappa**2 - 209938114493937457712064490143827601386506018476562500*kappa + 129212860735356614999409461513126646899712609619140625)/658392654944600256072817501262761652469635009765625",
"remaining_necessary_blend_set": "(kappa <= 9183350/7757187) & (275*CRootOf(18970220085333072146922654876672*x**4 + 983068858743131321871913065744*x**3 + 20545012001982889486061566108*x**2 - 84930319019487646264794864*x - 56547962754516449585533, 1) <= kappa)"
}
],
"all_real_blends_excluded": false,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 4,
"c_exact": "3/10",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "196441929*kappa/10000000000 + z**3 + z**2*(12889197*kappa/78125000 - 26554/15625) + z*(6577497/20000000 - 1427750991*kappa/6250000000) - 721143/32000000",
"discriminant": "81*(12569191132135997739991245645636*kappa**4 + 144984152605841412353954311972500*kappa**3 + 806119180763108513525782828515625*kappa**2 - 433252569373390008245689453125000*kappa - 698725608071733878916015625000000)/953674316406250000000000000000000000",
"remaining_necessary_blend_set": "((kappa <= 5564375/4850418) & (625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "312940017*kappa/400000000000 + z**4 + z**3*(12889197*kappa/78125000 - 69983/31250) + z**2*(46478961/50000000 - 4023928053*kappa/15625000000) + z*(144391355973*kappa/2500000000000 - 1046072151/8000000000) + 1712421/1280000000",
"discriminant": "531441*(13742162730699259662703900045146178948159496172155904*kappa**6 + 68689754458069299922529635667338386200327377088160000*kappa**5 + 827826580017715125525524879724515847576629433656250000*kappa**4 - 3636787125959024417306021160079796897744553046875000000*kappa**3 - 7454600075430618999018081662228208595360906982421875000*kappa**2 + 24095461123698772057877345549373956722352600097656250000*kappa - 13981690182850606444245638702830275204837322235107421875)/2384185791015625000000000000000000000000000000000000000000000000",
"remaining_necessary_blend_set": "(625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 1) <= kappa) & (kappa <= 625*CRootOf(13742162730699259662703900045146178948159496172155904*x**6 + 109903607132910879876047417067741417920523803341056*x**5 + 2119236044845350721345343692094760569796171350160*x**4 - 14896280067928164013285462671686848093161689280*x**3 - 48854467054342104671964899981578787850557240*x**2 + 252659222432435636093607954867803460440976*x - 234573836242764120046101037575343338451, 2))"
},
{
"scalar_harmonic_degree": 3,
"characteristic_polynomial": "59145663213*kappa/10000000000000 + z**4 + z**3*(12889197*kappa/78125000 - 92483/31250) + z**2*(103535919/50000000 - 4629995487*kappa/15625000000) + z*(1311828205869*kappa/12500000000000 - 20031078303/40000000000) + 621254529/32000000000",
"discriminant": "4782969*(84094875673931586649040468217464079700594065004056576*kappa**6 + 168749712147860454216124612065288397329371393303040000*kappa**5 + 2372005176671215636845971610641756985113222544156250000*kappa**4 - 2951391333693440926837794734019861571920183671875000000*kappa**3 - 704149155593370100404596770370411962821004266357421875000*kappa**2 + 1594156143839488002441069760546302822210134887695312500000*kappa - 896508828803876248871732166108019907094180583953857421875)/59604644775390625000000000000000000000000000000000000000000000000",
"remaining_necessary_blend_set": "(625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 1) <= kappa) & (kappa <= 625*CRootOf(13742162730699259662703900045146178948159496172155904*x**6 + 109903607132910879876047417067741417920523803341056*x**5 + 2119236044845350721345343692094760569796171350160*x**4 - 14896280067928164013285462671686848093161689280*x**3 - 48854467054342104671964899981578787850557240*x**2 + 252659222432435636093607954867803460440976*x - 234573836242764120046101037575343338451, 2))"
},
{
"scalar_harmonic_degree": 4,
"characteristic_polynomial": "15334060833*kappa/625000000000 + z**4 + z**3*(12889197*kappa/78125000 - 60304/15625) + z**2*(404389233/100000000 - 10775159559*kappa/31250000000) + z*(988154316567*kappa/6250000000000 - 30457879929/20000000000) + 257512689/2000000000",
"discriminant": "-26040609*(106951024097141207934267956679935318075361883367664*kappa**6 - 45981693244270747254641499883500911366815485095940000*kappa**5 + 934636015281693611921654414996972213979722824828125000*kappa**4 - 10132970047628713818195556253092680356991645380859375000*kappa**3 + 105633750474346521621569009271718419050976338958740234375*kappa**2 - 213973939610217576432758866583001403994758987426757812500*kappa + 118683069535045502257763882893570823471784591674804687500)/3725290298461914062500000000000000000000000000000000000000000000",
"remaining_necessary_blend_set": "(625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 1) <= kappa) & (kappa <= 625*CRootOf(13742162730699259662703900045146178948159496172155904*x**6 + 109903607132910879876047417067741417920523803341056*x**5 + 2119236044845350721345343692094760569796171350160*x**4 - 14896280067928164013285462671686848093161689280*x**3 - 48854467054342104671964899981578787850557240*x**2 + 252659222432435636093607954867803460440976*x - 234573836242764120046101037575343338451, 2))"
},
{
"scalar_harmonic_degree": 5,
"characteristic_polynomial": "938820051*kappa/12500000000 + z**4 + z**3*(12889197*kappa/78125000 - 77179/15625) + z**2*(721360107/100000000 - 12593361861*kappa/31250000000) + z*(5333650983*kappa/25000000000 - 314482257/80000000) + 22851963/40000000",
"discriminant": "-3720087*(5837291548694411502960809018954667362777298645456*kappa**6 - 302219606649175817204012889606645852872991840960000*kappa**5 + 5915014420619975863223998831175269768290021328125000*kappa**4 - 67843783943446622457033812193353002961449306640625000*kappa**3 + 477473655018477248182053600524680542272510528564453125*kappa**2 - 909945372152705392342143348520686753141975402832031250*kappa + 499794171265984294779381394765423735439777374267578125)/298023223876953125000000000000000000000000000000000000000000",
"remaining_necessary_blend_set": "(625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 1) <= kappa) & (kappa <= 625*CRootOf(13742162730699259662703900045146178948159496172155904*x**6 + 109903607132910879876047417067741417920523803341056*x**5 + 2119236044845350721345343692094760569796171350160*x**4 - 14896280067928164013285462671686848093161689280*x**3 - 48854467054342104671964899981578787850557240*x**2 + 252659222432435636093607954867803460440976*x - 234573836242764120046101037575343338451, 2))"
},
{
"scalar_harmonic_degree": 6,
"characteristic_polynomial": "76044424131*kappa/400000000000 + z**4 + z**3*(12889197*kappa/78125000 - 193733/31250) + z**2*(60086223/5000000 - 367864947*kappa/781250000) + z*(133003667871*kappa/500000000000 - 14346778077/1600000000) + 2520624663/1280000000",
"discriminant": "-33480783*(45338079683217801219044625518792396950743317362944*kappa**6 - 2822092455480676308824351518275639661430930150640000*kappa**5 + 57562874364226122194943225084982373725896818656250000*kappa**4 - 674868196783035168527051861002920419243136640625000000*kappa**3 + 3979844291752746692120882676162057770689324951171875000*kappa**2 - 7294015169976566815515041632060988444010162353515625000*kappa + 3977091029972636045761734718202140046894550323486328125)/3814697265625000000000000000000000000000000000000000000000000",
"remaining_necessary_blend_set": "(625*CRootOf(12569191132135997739991245645636*x**4 + 231974644169346259766326899156*x**3 + 2063665102753557794626004041*x**2 - 1774602524153405473774344*x - 4579168145058915148864, 1) <= kappa) & (kappa <= 625*CRootOf(13742162730699259662703900045146178948159496172155904*x**6 + 109903607132910879876047417067741417920523803341056*x**5 + 2119236044845350721345343692094760569796171350160*x**4 - 14896280067928164013285462671686848093161689280*x**3 - 48854467054342104671964899981578787850557240*x**2 + 252659222432435636093607954867803460440976*x - 234573836242764120046101037575343338451, 2))"
}
],
"all_real_blends_excluded": false,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 5,
"c_exact": "3/10",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "188586950949*kappa/20904541015625 + z**3 + z**2*(37347309*kappa/293265625 - 4597829/3853125) + z*(2287413679/13378906250 - 1116471926759*kappa/9164550781250) - 311116419/30517578125",
"discriminant": "(97224429699039559334743356228435270406041*kappa**4 + 998107249606508536115174343507027969377262*kappa**3 + 5623458389715327744102061772733072857730340*kappa**2 - 470721320359412991631704198914718916138150*kappa - 7620292295687380024771800061508487957715125)/585099468437378876900766044855117797851562500",
"remaining_necessary_blend_set": "((kappa <= 23679416335/20954105661) & (137*CRootOf(97224429699039559334743356228435270406041*x**4 + 7285454376689843329307841923408963280126*x**3 + 299614171757436610586715422917207781860*x**2 - 183063671288777928052548288358198550*x - 21631635758049453889486753124095125, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 137*CRootOf(97224429699039559334743356228435270406041*x**4 + 7285454376689843329307841923408963280126*x**3 + 299614171757436610586715422917207781860*x**2 - 183063671288777928052548288358198550*x - 21631635758049453889486753124095125, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "112708899576*kappa/476837158203125 + z**4 + z**3*(37347309*kappa/293265625 - 6151409/3853125) + z**2*(16478247984/33447265625 - 3198035621564*kappa/22911376953125) + z*(73890586807092*kappa/2863922119140625 - 230683961052/4180908203125) + 45020584728/95367431640625",
"discriminant": "64*(56491388267370445649149540176555437925149623776098487846039189763005*kappa**6 + 323081449835826239641844483378811671589693441270553954947730257570294*kappa**5 + 3177457550205593736077846792682776952814430620464010276257870345719675*kappa**4 - 5476010039632566939347838512229063527548058768246143280787520884643500*kappa**3 - 70262302068217162861502940895304592873033696617881915898573559346268125*kappa**2 + 149050441184201353107084284290836170537377440450650751670423191047793750*kappa - 77035163031817817012044959001509893305240355755923101333431367764946875)/46286796288787784559429780479290696300342933255933530745096504688262939453125",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 6,
"c_exact": "3/10",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "862962441*kappa/177625000000 + z**3 + z**2*(22143*kappa/214375 - 644573/710500) + z*(145570109/1421000000 - 6458063667*kappa/87036250000) - 15787707/2900000000",
"discriminant": "39*(71730251666879223754946077357824*kappa**4 + 688203325080348152282479917247680*kappa**3 + 3962018596418626530492959045839200*kappa**2 + 1147152159790308832411393187853500*kappa - 6948676585429104963781318438296875)/74952233310580245156250000000000000000",
"remaining_necessary_blend_set": "((kappa <= 1289329405/1150616588) & (35*CRootOf(71730251666879223754946077357824*x**4 + 19662952145152804350927997635648*x**3 + 3234300895035613494279966568032*x**2 + 26755735505313325537291969396*x - 4630521672922352329050441275, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 35*CRootOf(71730251666879223754946077357824*x**4 + 19662952145152804350927997635648*x**3 + 3234300895035613494279966568032*x**2 + 26755735505313325537291969396*x - 4630521672922352329050441275, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "2081079*kappa/22656250000 + z**4 + z**3*(22143*kappa/214375 - 871933/710500) + z**2*(425738601/1421000000 - 258226947*kappa/3001250000) + z*(2414470421*kappa/177625000000 - 567935769/20300000000) + 3697883/18125000000",
"discriminant": "(146126616283672666923930880872864954835004453357359104*kappa**6 + 1124305856169032265988245036830356888083935662171499520*kappa**5 + 7562505565810091966569652641678327949194317076422048000*kappa**4 + 12942060980224888153686462952665968297255279682446480000*kappa**3 - 309867620500449148063193395301323705672472783795540250000*kappa**2 + 560309477781699057256104859992349015055715641044910287500*kappa - 272517008831305560037382445691379513038901234972900515625)/37836655635572341202387851562500000000000000000000000000000000",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 10,
"c_exact": "3/10",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "2103971942561931*kappa/2321410996093750000 + z**3 + z**2*(98762105403*kappa/1702473781250 - 164509285121/371425759375) + z*(98232209922591/3714257593750000 - 1350851775854524281*kappa/68537338248671875000) - 126083944791/125804687500000",
"discriminant": "81*(104530713826708274955556643905013111467926523300664961168746648004*kappa**4 + 958205690247005905485273373712900166329035399421782048169464476660*kappa**3 + 5760848652758157807015136273937317123040544745458663647576497443025*kappa**2 + 8651364795235690797605757516973906540904279940886114161812006003500*kappa - 17591710260194526362374462636903846348037161149080357742719006397500)/14121762710841341978329494826037514475852513893151283264160156250000000000",
"remaining_necessary_blend_set": "((kappa <= 57446024475455/51949924507702) & (36905*CRootOf(104530713826708274955556643905013111467926523300664961168746648004*x**4 + 25964115709172358907608003623164887314158932378316814745141972*x**3 + 4229763121908692982269045056650059000020609528446051984961*x**2 + 172119072038473135173666053400791927018751910002567308*x - 9483468911008914506635088424256714146968473187516, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 36905*CRootOf(104530713826708274955556643905013111467926523300664961168746648004*x**4 + 25964115709172358907608003623164887314158932378316814745141972*x**3 + 4229763121908692982269045056650059000020609528446051984961*x**2 + 172119072038473135173666053400791927018751910002567308*x - 9483468911008914506635088424256714146968473187516, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "3613294738053*kappa/491424560546875000 + z**4 + z**3*(98762105403*kappa/1702473781250 - 228691656341/371425759375) + z**2*(298849577160843/3714257593750000 - 1614032430156657693*kappa/68537338248671875000) + z*(18713756887036085043*kappa/7788333891894531250000 - 1932688682242803/422074726562500000) + 4073250191967/196569824218750000",
"discriminant": "-6908733*(120769713906577267278260090591608002855431410762457660492087132782542096419384115951201541553526192*kappa**6 - 13356910279289949980988330086015145548183916635142875684139927663080508027813762543285646155909818880*kappa**5 + 34108495054246301188427949107316366675654543785604506859216732511699800949081625379000807662064593000*kappa**4 - 813023091720737811270097237870212874535921720980969514344980201004166330574922551642727663609038060000*kappa**3 + 4054625899520786680144503774733022944141415221549439025752476504831347731930843766783096203975851596875*kappa**2 - 5790506579097644657660704993424838190829232322222335135885377521287369562640481337348950442963120425000*kappa + 2528407686125553567038598164068070476427090570221821856519078339302484493087949819071564716844495750000)/3426399710038933371522416761226120392268033851539795764957864381660046092292759567499160766601562500000000000000000000",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 2,
"c_exact": "3/5",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "z**3 - 101454*z**2/15625 + 563841*z/78125 - 22485276/9765625",
"discriminant": "-1112477019438583036656/37252902984619140625",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 3,
"c_exact": "3/5",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "172187072*kappa/322265625 + z**3 + z**2*(1339356*kappa/1890625 - 21292/5625) + z*(5092336/1546875 - 208775104*kappa/141796875) - 3226048/3515625",
"discriminant": "16384*(116580359756477525204874483*kappa**4 - 96144007591099286104878900*kappa**3 + 853737953783556376388920000*kappa**2 - 640196829206409490665375000*kappa - 1212135519739107921770312500)/5821422645784914493560791015625",
"remaining_necessary_blend_set": "((kappa <= 13861925/8071269) & (275*CRootOf(116580359756477525204874483*x**4 - 349614573058542858563196*x**3 + 11289096909534629770432*x**2 - 30783318609474235464*x - 211943646645182452, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 275*CRootOf(116580359756477525204874483*x**4 - 349614573058542858563196*x**3 + 11289096909534629770432*x**2 - 30783318609474235464*x - 211943646645182452, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "31326208*kappa/48828125 + z**4 + z**3*(1339356*kappa/1890625 - 39292/5625) + z**2*(2438944/171875 - 313973344*kappa/141796875) + z*(4818117632*kappa/3544921875 - 12046336/1171875) + 8388608/5859375",
"discriminant": "-17179869184*(2537047095000921424703884640693629522869*kappa**6 - 121026184063448342922191543840076843217800*kappa**5 + 1456025133284724306582738157290120392493750*kappa**4 - 7228710507754248828395195270554146852500000*kappa**3 + 21411546191184822199309048249231759564453125*kappa**2 - 30120142802534168299078420528806673359375000*kappa + 15130700484137820189475820966178988281250000)/219464218314866752024272500420920550823211669921875",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 4,
"c_exact": "3/5",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "14014053*kappa/68359375 + z**3 + z**2*(5185251*kappa/9765625 - 544421/218750) + z*(12954951/8750000 - 136990323*kappa/195312500) - 144423/437500",
"discriminant": "81*(27503923011284578295611713916944*kappa**4 + 75916471658022800637672066885000*kappa**3 - 61739710369930396400313590234375*kappa**2 + 819376324573732059872228027343750*kappa - 1392130688303695884398040771484375)/139756593853235244750976562500000000",
"remaining_necessary_blend_set": "((kappa <= 1114375/692052) & (625*CRootOf(27503923011284578295611713916944*x**4 + 121466354652836481020275307016*x**3 - 158053658547021814784802791*x**2 + 3356165425454006517236646*x - 9123467678867101347991, 1) <= kappa)) | ((-oo < kappa) & (kappa <= 625*CRootOf(27503923011284578295611713916944*x**4 + 121466354652836481020275307016*x**3 - 158053658547021814784802791*x**2 + 3356165425454006517236646*x - 9123467678867101347991, 0)))"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "44955972*kappa/341796875 + z**4 + z**3*(5185251*kappa/9765625 - 1016921/218750) + z**2*(282485691/43750000 - 7604687061*kappa/6835937500) + z*(19071461097*kappa/34179687500 - 26377569/7812500) + 3129597/8750000",
"discriminant": "-531441*(1483942662958663351504466931362449854680129534463936*kappa**6 - 25648417133229952267886333872411530530046359674140000*kappa**5 + 173673152266582262770758466714793437037294179429687500*kappa**4 - 651610180472539678934389791227576236240162397460937500*kappa**3 + 1413954593705769299702794932644194145420458221435546875*kappa**2 - 1538549545060494006942456843452595398934745788574218750*kappa + 635439267118061072499182816814115349709987640380859375)/16327078578015630228037480264902114868164062500000000000000",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 5,
"c_exact": "3/5",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "2077563283776*kappa/20904541015625 + z**3 + z**2*(123495156*kappa/293265625 - 7077428/3853125) + z*(5521282928/6689453125 - 1841984381152*kappa/4582275390625) - 4720457664/30517578125",
"discriminant": "-262144*(575335269508896375910969827759642144*kappa**4 - 13400556942161356361355584661919189497*kappa**3 + 33295994798384487683620167563109063140*kappa**2 - 105329957454764438938132421785509854600*kappa + 117142274193403472801218118482690015500)/146274867109344719225191511213779449462890625",
"remaining_necessary_blend_set": "(kappa <= 5613738715/3606880701) & (137*CRootOf(575335269508896375910969827759642144*x**4 - 97814284249352966141281639868023281*x**3 + 1773988747316558563781776736273060*x**2 - 40962853974061297277399260928200*x + 332530421261575182379042735500, 0) <= kappa)"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "19317357674496*kappa/476837158203125 + z**4 + z**3*(123495156*kappa/293265625 - 13291748/3853125) + z**2*(120820027776/33447265625 - 15047346853184*kappa/22911376953125) + z*(789909327479808*kappa/2863922119140625 - 6171203008512/4180908203125) + 11873192067072/95367431640625",
"discriminant": "-68719476736*(2401451785072190500829681129513546163050334058744189525184650455*kappa**6 - 37554730484052982170862003640144029702098628194818461951564307796*kappa**5 + 233311325364324706648921624452114134659925759554164990155746640200*kappa**4 - 818865649241037704310936924518284848172654221203597658811262739750*kappa**3 + 1628227799705203878922536219915866098476039694782823189451637790000*kappa**2 - 1621698490422911261281601219947304100284159639766034572050050237500*kappa + 619045158018101904294883119445437733409672414531068268169473134375)/46286796288787784559429780479290696300342933255933530745096504688262939453125",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 6,
"c_exact": "3/5",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "153964536*kappa/2775390625 + z**3 + z**2*(521454*kappa/1500625 - 257132/177625) + z*(11566259/22203125 - 1400878482*kappa/5439765625) - 957732/11328125",
"discriminant": "-12*(6083861251874985081242393586228*kappa**4 - 48530233102471912714985037706500*kappa**3 + 134923279799625348990536973030825*kappa**2 - 339655327499707876779016599640250*kappa + 317878097689064485450467790563125)/56040335574622070505619049072265625",
"remaining_necessary_blend_set": "(kappa <= 19553695/12830378) & (245*CRootOf(6083861251874985081242393586228*x**4 - 198082584091722092714224643700*x**3 + 2247784753013333594178042033*x**2 - 23096181182990616275804578*x + 88225934651083910220101, 0) <= kappa)"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "22546944*kappa/1416015625 + z**4 + z**3*(521454*kappa/1500625 - 484492/177625) + z**2*(50696451/22203125 - 2334451698*kappa/5439765625) + z*(3006693512*kappa/19427734375 - 2093436/2734375) + 15124928/283203125",
"discriminant": "-64*(125658544800486614911554025207223525679543263063664*kappa**6 - 1867881697985897304028447485937469977280577236846400*kappa**5 + 11160947164400850599431606981542627887953112852319000*kappa**4 - 37967968538908217989698225280398322654127170676435000*kappa**3 + 72278171562859709014223480472801890803679534651559375*kappa**2 - 68670288131008988750404079485126507894469681099381250*kappa + 25076899302939676213670458486352686346855333489609375)/21151688457701632573020931240534991957247257232666015625",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
},
{
"depth": 10,
"c_exact": "3/5",
"harmonics": [
{
"scalar_harmonic_degree": 1,
"characteristic_polynomial": "1614600981989784*kappa/145088187255859375 + z**3 + z**2*(171816419466*kappa/851236890625 - 285602232176/371425759375) + z*(35372969893269/232141099609375 - 656901219832013682*kappa/8567167281083984375) - 64138310028/3931396484375",
"discriminant": "-324*(5385188249306966881600061283848570495275841581428170625825031836*kappa**4 - 27502069498460825082874738713558389540626366903098647938024786740*kappa**3 + 83963400725178494644270384640997194700702099432000588153844357475*kappa**2 - 167279809668905874176463079701725585547518048181604622653575502750*kappa + 122214560001656252092724645853431618298901726114155281241481019375)/13790783897305998025712397291052260230324720598780550062656402587890625",
"remaining_necessary_blend_set": "(kappa <= 7305630653035/4983336364166) & (36905*CRootOf(5385188249306966881600061283848570495275841581428170625825031836*x**4 - 745212559232104730602214841174864911004643460319703236364308*x**3 + 61648086485897603778641440268140102011264159836823175859*x**2 - 3328035089543360054684194103417337164965308201817742*x + 65884326373365611547382850116460464198865386511, 0) <= kappa)"
},
{
"scalar_harmonic_degree": 2,
"characteristic_polynomial": "79810474715136*kappa/61428070068359375 + z**4 + z**3*(171816419466*kappa/851236890625 - 542331717056/371425759375) + z**2*(155674374309837/232141099609375 - 1143614090459204226*kappa/8567167281083984375) + z*(15014189788035721992*kappa/486770868243408203125 - 1729309994343324/13189835205078125) + 65862374691168/12285614013671875",
"discriminant": "-34012224*(61335091162503179653995902047711148977391796673153696821277100604280499598390327904119845743592304*kappa**6 - 840188970766141008809845897650394170784814361112802282696666647151540068215255542454994386164487840*kappa**5 + 4773222129584185079675247845969867595139699943476291462384890872914320488399667522356023671021011000*kappa**4 - 15622736578377545594984368734193499623651689376784797348690049006097253730725461486788081977001430000*kappa**3 + 27936861867287017241463016876180099436222547499537098203881918031343100238076651403281004983268709375*kappa**2 - 24645670537344944993421306024891464155287087766439638885680708769529296477740615353437576710585056250*kappa + 8358522732856889627176470042164172629448728185415798615768465746780726880478143573854333733176109375)/3267669401205952998659531365610237495677980281390949025113930112514539806645164077281151548959314823150634765625",
"remaining_necessary_blend_set": "False"
}
],
"all_real_blends_excluded": true,
"classification": "exact necessary root inequalities; nonempty sets do not establish stability"
}
],
"limits": [
"unrestricted real blend coefficient in the stated two-action family",
"exact ground-shell reducing harmonic blocks, not assigned excited electron states",
"no theorem for every local Lorentz scalar potential is assumed",
"nonempty necessary sets do not prove real-rootedness of quartics or full spatial stability"
]
}