files / problems / simon-ionization-excess / compute / q2 / certs / envelopes.json
{
"not_a_new_bound": true,
"is_new_bound": false,
"record": [
"Lieb Phys. Rev. A 29 (1984): Nc < 2Z+1",
"Nam arXiv:1009.2367v3: Nc < 1.22 Z + 3 Z^{1/3}",
"HPS arXiv:2504.18487v1 Prop. 2.4\u20132.5",
"q1 remainder tightening 2.953 / 3.892 / 3.9781 (same HPS chain)"
],
"constants": {
"b2": "1.207106781186547524400844362104849039285",
"b3": "1.118433799202706636641762578621416983973"
},
"at": [
{
"Z": 2,
"lieb": {
"U": "5.000000000000000000000000000000000000000",
"U_float": 5.0,
"max_integer_Nc": 4,
"excludes_N_geq": 5
},
"nam": {
"U": "6.219763149684619494301631821834685051711",
"U_float": 6.21976314968462,
"max_integer_Nc": 6,
"excludes_N_geq": 7
},
"hps_s2_printed": {
"U": "6.143579870061919616512632121753253996258",
"U_float": 6.143579870061919,
"max_integer_Nc": 6,
"excludes_N_geq": 7
},
"hps_s2_q1": {
"U": "6.134760422712655504359261647502306397804",
"U_float": 6.134760422712655,
"max_integer_Nc": 6,
"excludes_N_geq": 7
},
"hps_s3_printed": null,
"hps_s3_q1": null,
"hps_simplified_printed": null,
"hps_simplified_q1": null,
"best_published": {
"name": "lieb",
"U": "5.000000000000000000000000000000000000000",
"max_integer_Nc": 4,
"excludes_N_geq": 5
},
"zhislin_N0_at_least": 2,
"unsettled_integers": [
2,
3,
4
]
},
{
"Z": 3,
"lieb": {
"U": "7.000000000000000000000000000000000000000",
"U_float": 7.0,
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},
"nam": {
"U": "7.986748710922225146964914932340328765176",
"U_float": 7.9867487109222255,
"max_integer_Nc": 7,
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},
"hps_s2_printed": {
"U": "7.890379071669571384874582486223671499494",
"U_float": 7.890379071669571,
"max_integer_Nc": 7,
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},
"hps_s2_q1": {
"U": "7.880283324677419526198331018048210732376",
"U_float": 7.8802833246774195,
"max_integer_Nc": 7,
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},
"hps_s3_printed": null,
"hps_s3_q1": null,
"hps_simplified_printed": null,
"hps_simplified_q1": null,
"best_published": {
"name": "lieb",
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},
"zhislin_N0_at_least": 3,
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3,
4,
5,
6
]
},
{
"Z": 4,
"lieb": {
"U": "9.000000000000000000000000000000000000000",
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},
"nam": {
"U": "9.642203155904598424255116917816924781174",
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},
"hps_s2_printed": {
"U": "9.527134238572060542868426140665428607898",
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},
"hps_s2_q1": {
"U": "9.516022431208283146545164201190522450075",
"U_float": 9.516022431208283,
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},
"hps_s3_printed": {
"U": "10.80169030123177700668356904114970147015",
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},
"hps_s3_q1": {
"U": "10.78899109281603141088555539603552300406",
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},
"hps_simplified_printed": {
"U": "10.82360420787279789900682255708923304157",
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"excludes_N_geq": 11
},
"hps_simplified_q1": {
"U": "10.78884012483469433050976020358916949066",
"U_float": 10.788840124834694,
"max_integer_Nc": 10,
"excludes_N_geq": 11
},
"best_published": {
"name": "lieb",
"U": "9.000000000000000000000000000000000000000",
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},
"zhislin_N0_at_least": 4,
"unsettled_integers": [
4,
5,
6,
7,
8
]
},
{
"Z": 5,
"lieb": {
"U": "11.00000000000000000000000000000000000000",
"U_float": 11.0,
"max_integer_Nc": 10,
"excludes_N_geq": 11
},
"nam": {
"U": "11.22992784003009096805932661763158032960",
"U_float": 11.229927840030092,
"max_integer_Nc": 11,
"excludes_N_geq": 12
},
"hps_s2_printed": {
"U": "11.09706270809576071048942407325407112163",
"U_float": 11.097062708095761,
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},
"hps_s2_q1": {
"U": "11.08509287646902383156395231114626410086",
"U_float": 11.085092876469023,
"max_integer_Nc": 11,
"excludes_N_geq": 12
},
"hps_s3_printed": {
"U": "12.38878214652985756509485145766158683823",
"U_float": 12.388782146529858,
"max_integer_Nc": 12,
"excludes_N_geq": 13
},
"hps_s3_q1": {
"U": "12.37510233895644398918002658668123595735",
"U_float": 12.375102338956443,
"max_integer_Nc": 12,
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},
"hps_simplified_printed": {
"U": "12.43240378670678795741243549017544043947",
"U_float": 12.432403786706788,
"max_integer_Nc": 12,
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},
"hps_simplified_q1": {
"U": "12.39495531347456829334560240586672990307",
"U_float": 12.394955313474568,
"max_integer_Nc": 12,
"excludes_N_geq": 13
},
"best_published": {
"name": "lieb",
"U": "11.00000000000000000000000000000000000000",
"max_integer_Nc": 10,
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},
"zhislin_N0_at_least": 5,
"unsettled_integers": [
5,
6,
7,
8,
9,
10
]
}
],
"punch": "At Z=2,3,4,5 the best published integer bound is Lieb. Nam and both HPS envelopes sit above 2Z+1 here, so they exclude no extra integer N. Unsettled: Z=2 has N=2,3,4; Z=3 has N=3,4,5,6; Z=4 has N=4..8; Z=5 has N=5..10. Hydrogen uniqueness N0(1)=2 is Lieb 1984, not claimed here."
}