Modular bootstrap

In 2019, Hartman, Mazáč, and Rastelli [3] discovered a remarkable equivalence between the spinless modular bootstrap bound for the spectral gap of conformal field theories with current algebra U(1)c×U(1)c (i.e., theories of c free bosons) and the linear programming bound for sphere packing density in dimension 2c. This bound is sharp for the E8 root lattice and the Leech lattice [4,2].

Under this equivalence, the scaling dimensions Δn of the conformal field theory are given by half the squared distances between points in the sphere packing and the degeneracies dn are the corresponding theta series coefficients.

Other than in dimensions 1, 8, and 24, this bound has not been exactly optimized. It is conjectured to match the densest sphere packing in dimension 2, while other cases are a mystery. The paper [1] computes the optimum numerically to high precision, and this web page displays part of the full data set. The tables below show the scaling dimensions and degeneracies for hypothetical conformal field theories achieving the bound. (To obtain odd sphere packing dimensions, we allow c to be a half-integer, in which case the CFT must be chiral.)

I believe that all the digits presented here are accurate, aside from rounding the last digit, except in the case of dimension 3, which is much more challenging numerically for unclear reasons. In that case Δ1 is an upper bound for the spectral gap, but it is unclear how close to the optimum we have come.

The scaling dimensions and degeneracies show numerous patterns that have yet to be explained (see [1, Section 4]). Can they somehow be characterized exactly, perhaps in terms of automorphic forms?

References

  1. N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat, and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, J. High Energy Phys. (2020), Paper No. 66, 44 pp., doi:10.1007/jhep12(2020)066, arXiv:2006.02560.
  2. H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), 1017-1033, doi:10.4007/annals.2017.185.3.8, arXiv:1603.06518.
  3. T. Hartman, D. Mazáč, and L. Rastelli, Sphere packing and quantum gravity, J. High Energy Phys. (2019), Paper No. 48, 66 pp., doi:10.1007/JHEP12(2019)048, arXiv:1905.01319.
  4. M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), 991-1015, doi:10.4007/annals.2017.185.3.7, arXiv:1603.04246.

Dimension 1

n Scaling dimension Δn Degeneracy dn
1 0.500000000000 2.0000
2 2.000000000000 2.0000
3 4.500000000000 2.0000
4 8.000000000000 2.0000
5 12.500000000000 2.0000
6 18.000000000000 2.0000
7 24.500000000000 2.0000
8 32.000000000000 2.0000
9 40.500000000000 2.0000
10 50.000000000000 2.0000
11 60.500000000000 2.0000
12 72.000000000000 2.0000
13 84.500000000000 2.0000
14 98.000000000000 2.0000
15 112.500000000000 2.0000

Dimension 2

n Scaling dimension Δn Degeneracy dn
1 0.577350269190 6.0000
2 1.732050807569 6.0000
3 2.309401076759 6.0000
4 4.041451884327 12.0000
5 5.196152422707 6.0000
6 6.928203230276 6.0000
7 7.505553499465 12.0000
8 9.237604307034 6.0000
9 10.969655114603 12.0000
10 12.124355652982 12.0000
11 14.433756729741 6.0000
12 15.588457268120 6.0000
13 16.165807537310 12.0000
14 17.897858344878 12.0000

Dimension 3

n Scaling dimension Δn Degeneracy dn
1 0.652038604084 13.4018
2 1.749909380418 24.8614
3 2.514640815146 23.7455
4 3.865125850442 47.6248
5 4.456191053793 23.0409
6 5.570309782339 41.0548
7 6.827051035245 64.4510
8 7.585504096334 38.5651
9 8.337857048838 32.8679
10 9.786857841408 79.9850
11 10.566078989360 43.5815
12 11.168623110808 32.7852
13 11.774456230127 39.1912
14 12.496435590785 45.7352
15 13.382419122342 51.1772

Dimension 4

n Scaling dimension Δn Degeneracy dn
1 0.724575648447 26.4302
2 1.786743532669 75.7139
3 2.660501420369 89.1337
4 3.839787768752 194.2921
5 4.629238766583 124.3223
6 5.739214724958 253.3545
7 6.795587623716 311.1023
8 7.732454260043 252.8666
9 8.609054459161 270.4661
10 9.856544494859 613.6197
11 10.673227076942 252.7442
12 11.753833785031 474.6704
13 12.693697947788 466.2060
14 13.701460922165 555.1642
15 14.812721681468 823.2113

Dimension 5

n Scaling dimension Δn Degeneracy dn
1 0.795367272698 48.5767
2 1.833339375556 197.5092
3 2.769101852829 302.0124
4 3.856506720124 676.3753
5 4.756184134560 586.7003
6 5.823494473118 1217.1148
7 6.816239227355 1313.5583
8 7.821561716476 1536.6110
9 8.760175645322 1737.1171
10 9.862016364118 3085.8336
11 10.746012000123 1697.4011
12 11.834737748848 3495.6842
13 12.798350324566 3285.3816
14 13.813014793307 3976.7058
15 14.788237497193 4182.7772

Dimension 6

n Scaling dimension Δn Degeneracy dn
1 0.864721288837 85.2621
2 1.885680671483 467.0991
3 2.856119698789 923.2615
4 3.894252020927 2132.5634
5 4.852182538423 2434.1515
6 5.884484699003 4925.4351
7 6.867763392120 5403.1184
8 7.885155853240 7868.9727
9 8.856041493877 9171.9763
10 9.893583771293 14206.2662
11 10.847442835673 11148.3548
12 11.889592640213 20596.0809
13 12.866547856577 19233.0958
14 13.880814833853 24579.0526
15 14.855310483034 25390.9126

Dimension 7

n Scaling dimension Δn Degeneracy dn
1 0.932871948832 144.8267
2 1.941647755354 1030.8824
3 2.931262731499 2583.3087
4 3.943776319695 6277.3480
5 4.930610078168 8998.1218
6 5.941684052623 17901.4294
7 6.932759933982 21618.2718
8 7.942476586752 35000.4892
9 8.931524540325 42577.9628
10 9.942764890238 63290.4925
11 10.929962747812 63765.8805
12 11.943226716667 105022.9315
13 12.932843708428 103350.9777
14 13.940909966279 140543.4581
15 14.930990496605 152871.3624

Dimension 8

n Scaling dimension Δn Degeneracy dn
1 1.000000000000 240.0000
2 2.000000000000 2160.0000
3 3.000000000000 6720.0000
4 4.000000000000 17520.0000
5 5.000000000000 30240.0000
6 6.000000000000 60480.0000
7 7.000000000000 82560.0000
8 8.000000000000 140400.0000
9 9.000000000000 181680.0000
10 10.000000000000 272160.0000
11 11.000000000000 319680.0000
12 12.000000000000 490560.0000
13 13.000000000000 527520.0000
14 14.000000000000 743040.0000
15 15.000000000000 846720.0000

Dimension 9

n Scaling dimension Δn Degeneracy dn
1 1.066247148837 390.0761
2 2.059961093056 4345.1572
3 3.065304683606 16467.2642
4 4.059910320630 46760.2894
5 5.065030210528 94113.9547
6 6.060155499784 193425.4888
7 7.065659175129 297978.5313
8 8.059312629843 522181.5488
9 9.065498356323 727601.3807
10 10.060306031363 1118129.3324
11 11.064956220708 1450025.8259
12 12.059794412001 2161957.1255
13 13.065665774074 2552777.3314
14 14.059750223809 3642586.1934
15 15.065360648423 4340318.4045

Dimension 10

n Scaling dimension Δn Degeneracy dn
1 1.131726299211 624.1171
2 2.121022129771 8457.3899
3 3.128789125093 38409.2555
4 4.121724249192 119984.0666
5 5.128087140569 275219.6400
6 6.121746969216 591383.2623
7 7.129456854615 1016157.9508
8 8.120375264698 1833500.5528
9 9.129277299081 2759768.6430
10 10.121858615823 4374625.6819
11 11.128123541221 6104038.1020
12 12.121476161280 9100850.4785
13 13.129384120631 11661964.5504
14 14.120729825648 16746731.8190
15 15.129094373888 20924851.6314

Dimension 11

n Scaling dimension Δn Degeneracy dn
1 1.196528819622 985.6577
2 2.182838026955 16017.4995
3 3.191336461111 85963.3207
4 4.184425039584 297140.9683
5 5.190337442751 764546.6278
6 6.184205905953 1738421.1646
7 7.192026332083 3289006.0683
8 8.182679631702 6146017.3350
9 9.191971654335 9964559.7364
10 10.184082357000 16336520.0463
11 11.190651326266 24237613.8132
12 12.184063281875 36757999.7099
13 13.191843338791 50381503.2164
14 14.182866335545 73054698.3078
15 15.191879034486 95815488.5160

Dimension 12

n Scaling dimension Δn Degeneracy dn
1 1.260729735848 1539.5887
2 2.245168717551 29643.0183
3 3.253434963410 185775.5197
4 4.247463316686 712498.6599
5 5.252328081628 2034113.8823
6 6.247089093589 4931819.4840
7 7.253978427978 10163185.7318
8 8.245685879532 19802776.4511
9 9.254050667428 34395888.0636
10 10.246737846156 58476733.3293
11 11.252899931934 91684712.6544
12 12.246993986448 142769910.3496
13 13.253742083856 206912243.3764
14 14.245767212922 304963162.8014
15 15.254053839255 418948593.5413

Dimension 13

n Scaling dimension Δn Degeneracy dn
1 1.324391476210 2382.1984
2 2.307843800283 53782.9347
3 3.315354699712 389603.4159
4 4.310563009643 1658868.7270
5 5.314298799277 5215213.7118
6 6.310120091190 13542014.5533
7 7.315713149795 30149498.0094
8 8.308982269872 61598886.7157
9 9.315857483703 113976895.6750
10 10.309652611691 201486964.2941
11 11.315003959437 332507226.8842
12 12.310014982515 534378877.0270
13 13.315494074155 812542838.5923
14 14.309018229786 1224833642.8499
15 15.315907206291 1755831782.6128

Dimension 14

n Scaling dimension Δn Degeneracy dn
1 1.387566606866 3655.7809
2 2.370740354929 95915.5463
3 3.377242676607 796059.2590
4 4.373600506765 3759700.1953
5 5.376340858297 12946519.9750
6 6.373151755476 36079811.0559
7 7.377448607789 86279308.0622
8 8.372312777926 185551965.6174
9 9.377611741906 363995121.1903
10 10.372682270420 670676061.0395
11 11.377045838861 1161363935.0925
12 12.373026123354 1931933184.6574
13 13.377287511975 3066755355.7930
14 14.372330337982 4749609749.6948
15 15.377666856235 7077866477.0113

Dimension 15

n Scaling dimension Δn Degeneracy dn
1 1.450299859156 5569.8306
2 2.433768550885 168486.8584
3 3.439175414989 1589858.5719
4 4.436532838943 8314079.1619
5 5.438476935802 31234425.5709
6 6.436118560746 93482410.7493
7 7.439284099158 239135897.3719
8 8.435547427980 542566152.7292
9 9.439432645388 1124311599.7227
10 10.435719937225 2163000420.9000
11 11.439088794290 3920516807.7848
12 12.435987967044 6762258088.9933
13 13.439188281155 11171224401.6387
14 14.435553941970 17827847196.0744
15 15.439474750525 27532613756.0868

Dimension 16

n Scaling dimension Δn Degeneracy dn
1 1.512629655841 8431.7163
2 2.496862059014 292026.0935
3 3.501188743278 3111809.1445
4 4.499356769355 17976615.3339
5 5.500700822965 73453050.3608
6 6.499000100254 236038222.7312
7 7.501251027920 644048112.9629
8 8.498639002092 1543252274.4660
9 9.501370436096 3369293779.5773
10 10.498700631625 6775380546.1015
11 11.501176403749 12829695127.5041
12 12.498883782080 22968181925.2368
13 13.501207762340 39406853585.8334
14 14.498636709209 64911931898.9957
15 15.501396842832 103663759947.2294

Dimension 17

n Scaling dimension Δn Degeneracy dn
1 1.574589277869 12690.9757
2 2.559971518684 500122.0004
3 3.563295044487 5982249.7723
4 4.562086410525 38076692.1294
5 5.562996579735 168789400.4403
6 6.561798022476 581904296.0420
7 7.563346222437 1690050404.5968
8 8.561586232562 4277931694.7046
9 9.563434224840 9822186036.3116
10 10.561594993516 20655929040.6427
11 11.563331809042 40803260414.4221
12 12.561707699299 75856512918.6674
13 13.563336552887 134984465919.6214
14 14.561578009461 229703526353.6277
15 15.563448625360 378786969928.1335

Dimension 18

n Scaling dimension Δn Degeneracy dn
1 1.636207769259 19003.1357
2 2.623059998669 847325.8476
3 3.625493403343 11316588.0758
4 4.624741523534 79142535.7983
5 5.625347219938 379777604.7477
6 6.624523061416 1403127537.6560
7 7.625551594571 4330766775.0393
8 8.624408793954 11576940827.3652
9 9.625612090964 27917694048.1091
10 10.624398181608 61401908817.8177
11 11.625561323947 126397510786.5781
12 12.624461214944 244061248082.4853
13 13.625558713336 450019466793.5992
14 14.624398299893 791402947736.3826
15 15.625619032594 1346321500636.5312

Dimension 19

n Scaling dimension Δn Degeneracy dn
1 1.697510648238 28321.4824
2 2.686099788120 1421635.8605
3 3.687775639022 21098222.8746
4 4.687341780752 161668680.1142
5 5.687738154875 838147625.2893
6 6.687188415743 3314442655.5520
7 7.687844815406 10857703147.0477
8 8.687132710767 30634366563.2380
9 9.687883800763 77516629559.9564
10 10.687119180620 178260994580.4329
11 11.687860106170 382104376778.8927
12 12.687151097406 766253017000.8087
13 13.687857200494 1463058223074.6471
14 14.687123062066 2659046938706.7805
15 15.687886679711 4663864132807.5660

Dimension 20

n Scaling dimension Δn Degeneracy dn
1 1.758520472847 42028.7657
2 2.749070092205 2364121.9825
3 3.750129906657 38818365.9614
4 4.749904254907 325015073.9176
5 5.750158119357 1817081851.2401
6 6.749806725062 7681017131.3202
7 7.750204560441 26676313725.7074
8 8.749782997289 79380789105.9521
9 9.750227953848 210610329159.7587
10 10.749772846229 506190236568.6063
11 11.750217524643 1129156959541.0439
12 12.749787205690 2351096055442.9899
13 13.750215911968 4646315624273.7588
14 14.749775939520 8725839697489.9737
15 15.750228837691 15773507180775.2201

Dimension 21

n Scaling dimension Δn Degeneracy dn
1 1.819257295700 62125.8754
2 2.811955353901 3899633.3513
3 3.812542840164 70565290.1847
4 4.812442588734 643838751.2216
5 5.812598985903 3874946663.0661
6 6.812388925084 17485961132.8520
7 7.812612680460 64319532404.8449
8 8.812380648973 201695374703.9243
9 9.812625440869 560745703093.1158
10 10.812374958936 1407807029326.2817
11 11.812621128888 3266598357766.8267
12 12.812380449340 7059757207322.0347
13 13.812620540876 14434989934812.7323
14 14.812376534436 28005340615449.4547
15 15.812625465662 52160803841896.3879

Dimension 22

n Scaling dimension Δn Degeneracy dn
1 1.879739033245 91501.6671
2 2.874744016162 6384641.3351
3 3.875000810170 126865395.2329
4 4.874966992080 1258130292.4893
5 5.875055172641 8137717142.9445
6 6.874944004754 39150435207.0381
7 7.875054750664 152380818261.5732
8 8.874941959117 503132371340.9894
9 9.875060700971 1464908986817.9242
10 10.874939654698 3839568264756.7685
11 11.875059077669 9263451571177.6343
12 12.874941257897 20771762661616.1219
13 13.875058959748 43929423611018.0942
14 14.874940195902 88018659717256.9435
15 15.875060466053 168875597254049.0979

Dimension 23

n Scaling dimension Δn Degeneracy dn
1 1.939981768308 134318.2129
2 2.937427598354 10381480.7601
3 3.937490641514 225774253.4669
4 4.937484606854 2427609261.5744
5 5.937522980106 16847623451.4586
6 6.937479147476 86302890718.3005
7 7.937519844671 355115567962.3011
8 8.937478917289 1233556232187.9055
9 9.937521804435 3759322585599.2702
10 10.937478423123 10280710157740.7658
11 11.937521328762 25780476002595.3542
12 12.937478676514 59953795666202.8985
13 13.937521329105 131109961005020.9847
14 14.937478515898 271212274348265.9619
15 15.937521617452 535929617386133.8710

Dimension 24

n Scaling dimension Δn Degeneracy dn
1 2.000000000000 196560.0000
2 3.000000000000 16773120.0000
3 4.000000000000 398034000.0000
4 5.000000000000 4629381120.0000
5 6.000000000000 34417656000.0000
6 7.000000000000 187489935360.0000
7 8.000000000000 814879774800.0000
8 9.000000000000 2975551488000.0000
9 10.000000000000 9486551299680.0000
10 11.000000000000 27052945920000.0000
11 12.000000000000 70486236999360.0000
12 13.000000000000 169931095326720.0000
13 14.000000000000 384163586352000.0000
14 15.000000000000 820166620815360.0000
15 16.000000000000 1668890090322000.0000

Dimension 25

n Scaling dimension Δn Degeneracy dn
1 2.059806851746 286815.6936
2 3.062456970954 26939953.8347
3 4.062517577419 695628275.1846
4 5.062515665363 8731845733.2943
5 6.062484642906 69438192458.9121
6 7.062510952696 401770764471.3965
7 8.062489615212 1842862074141.0864
8 9.062511008562 7068224659759.6681
9 10.062489089839 23561901889060.1643
10 11.062510691206 70027344472844.0957
11 12.062489259264 189506625614947.8761
12 13.062510776741 473431132947112.1316
13 14.062489271793 1106148049420333.4404
14 15.062510718976 2436530220343840.2428
15 16.062489308602 5104404041067761.0718

Dimension 26

n Scaling dimension Δn Degeneracy dn
1 2.119414245636 417389.3795
2 3.124795704833 43031222.0707
3 4.125033154887 1205895353.3507
4 5.125032480368 16302052765.1657
5 6.124975787088 138459801976.7024
6 7.125015385861 849916694827.2334
7 8.124984890466 4110771141062.7998
8 9.125015804850 16548370663894.7941
9 10.124984816250 57647842393112.9029
10 11.125014848610 178467329462141.5303
11 12.124984998849 501448128589984.8223
12 13.125015018432 1297624960202082.3655
13 14.124985071203 3132610510239279.5568
14 15.125014880400 7117092615989578.7759
15 16.124985218482 15347635386548681.4548

Dimension 27

n Scaling dimension Δn Degeneracy dn
1 2.178833049525 605878.0311
2 3.187014524658 68380376.8319
3 4.187537594352 2074690879.5982
4 5.187550097574 30145045572.2734
5 6.187472532140 273061844709.0562
6 7.187515870335 1776199690041.1412
7 8.187483354037 9051281169439.0931
8 9.187516898359 38215427350348.0300
9 10.187484338014 139047443831568.7066
10 11.187515349076 448159953469579.5757
11 12.187484451902 1306930751135141.5222
12 13.187515492322 3501868297689205.3247
13 14.187484611322 8732597918285572.9841
14 15.187515306130 20457268695769650.1949
15 16.187484856325 45401197838132475.9507

Dimension 28

n Scaling dimension Δn Degeneracy dn
1 2.238073201878 877407.9284
2 3.249112638147 108138827.8410
3 4.250022789386 3544218607.7996
4 5.250067268284 55243972012.1743
5 6.249974036015 532951726149.1517
6 7.250014324967 3669590164575.0201
7 8.249983482639 19685726900605.1145
8 9.250015886706 87110332070036.7539
9 10.249985816960 330871678254674.1289
10 11.250014009511 1109706445627116.7412
11 12.249985834768 3357531159390103.4611
12 13.250014018586 9311740065995990.5142
13 14.249986072730 23979749197314636.4485
14 15.250013817370 57907157861036116.5667
15 16.249986382686 132234656811776814.8897

Dimension 29

n Scaling dimension Δn Degeneracy dn
1 2.297143818336 1267801.4017
2 3.311089946073 170240111.1781
3 4.312481594618 6014534570.8230
4 5.312582103352 100388432525.5856
5 6.312479414725 1030050396545.8900
6 7.312512138393 7499334853027.7191
7 8.312484407684 42317715956650.8051
8 9.312513771341 196124714003914.3188
9 10.312488107806 777245583719659.1165
10 11.312511917131 2711293611832295.4843
11 12.312488050728 8507858011004073.8066
12 13.312511732487 24414042285442244.8521
13 14.312488330528 64909124966894200.7519
14 15.312511535203 161531386423043022.4927
15 16.312488673796 379466837896950889.8458

Dimension 30

n Scaling dimension Δn Degeneracy dn
1 2.356053283145 1828055.6208
2 3.372946891381 266862766.9796
3 4.374907745733 10143104904.1159
4 5.375092279402 180978119370.6788
5 6.374987687103 1972451145152.8634
6 7.375010263207 15168934470343.1337
7 8.374985693951 89965510309400.8808
8 9.375011175178 436404223145097.5399
9 10.374990533705 1803529103080315.3103
10 11.375009677472 6540467503399862.6293
11 12.374990462712 21277638438557274.8823
12 13.375009296098 63154096824308413.6438
13 14.374990736042 173300983189215924.1309
14 15.375009107519 444324129982113802.2045
15 16.374991086013 1073568328057041406.8888

Dimension 31

n Scaling dimension Δn Degeneracy dn
1 2.414809328000 2630669.8857
2 3.434684339818 416644125.6710
3 4.437295777285 17005298296.9932
4 5.437595199352 323822829273.2948
5 6.437497750675 3744088142291.8216
6 7.437509289553 30383876267187.0208
7 8.437487176385 189255205606844.3745
8 9.437508487111 960245223794961.6163
9 10.437492728066 4136178953197849.0967
10 11.437507592413 15586742266730702.6842
11 12.437492728650 52550964595318020.8494
12 13.437507058884 161276684736875173.8262
13 14.437492951798 456651949976311051.6535
14 15.437506874615 1205922624538000093.4519
15 16.437493286192 2996196370483018112.6978

Dimension 32

n Scaling dimension Δn Degeneracy dn
1 2.473419100338 3778574.6399
2 3.496303485126 648022165.0090
3 4.499640942569 28352096172.5803
4 5.500088115554 575316472691.1290
5 6.500008378053 7048164890529.0025
6 7.500009503681 60297389364059.5488
7 8.499988842139 394144574759065.5852
8 9.500005954204 2090436469436921.5868
9 10.499994525224 9380254667209781.7785
10 11.500005783060 36715453857476778.7677
11 12.499994687761 128240670800707207.1353
12 13.500005173636 406805836361623140.3947
13 14.499994832837 1188217877975336955.3199
14 15.500004985872 3231140501694212926.5242
15 16.499995132983 8253450010963973061.3425

Dimension 33

n Scaling dimension Δn Degeneracy dn
1 2.531889222761 5417718.8673
2 3.557805773473 1004268344.0385
3 4.561939137943 47022369187.6584
4 5.562568222887 1015286034860.2670
5 6.562518226134 13163741588689.6943
6 7.562510935563 118608834317198.5789
7 8.562490747096 813021255868124.4599
8 9.562503737733 4504665857709947.0880
9 10.562495885913 21046574580579684.4536
10 11.562504271345 85527549124963291.5248
11 12.562496285338 309368987095202086.7502
12 13.562503676643 1014073519105866282.7445
13 14.562496345304 3054604923758787655.4987
14 15.562503478982 8551320676603088744.5174
15 16.562496596388 22451719905092192377.9423

Dimension 34

n Scaling dimension Δn Degeneracy dn
1 2.590225844945 7754796.9733
2 3.619192843042 1551049913.4617
3 4.624186832797 77600083373.4565
4 5.625032727991 1780342079042.1187
5 6.625025852632 24402003085097.8454
6 7.625013398492 231355984658305.8895
7 8.624992958236 1661788378549029.1822
8 9.625001945104 9612887887254966.4694
9 10.624996846868 46740995566373592.6996
10 11.625003031938 197119151514230755.0679
11 12.624997525099 738137965672334646.1489
12 13.625002540956 2499337696257065043.5965
13 14.624997513274 7761928755773234321.3041
14 15.625002331645 22364590168984997855.4151
15 16.624997705198 60342373842533177823.7481

Dimension 35

n Scaling dimension Δn Degeneracy dn
1 2.648434689160 11082189.4814
2 3.680466475632 2387768401.8286
3 4.686381005631 127458397501.7055
4 5.687478899733 3103068427229.7554
5 6.687529736155 44912960209709.3823
6 7.687516522712 447670932020159.5928
7 8.687495515165 3367078561419166.2222
8 9.687500646261 20323234048265222.0033
9 10.687497486855 102789720327291089.9180
10 11.687502023478 449682759851024452.0994
11 12.687498440320 1742605229178501778.0013
12 13.687501710316 6093221817259043480.0970
13 14.687498385897 19504478014252119564.8696
14 15.687501493794 57827614013316454527.2445
15 16.687498514159 160306323251437736987.4184

Dimension 36

n Scaling dimension Δn Degeneracy dn
1 2.706521090344 15813020.4496
2 3.741628557816 3664524584.2439
3 4.748519086264 208413040443.8186
4 5.749904105061 5377506236535.4722
5 6.750028297354 82103945849354.4664
6 7.750019784449 859625443625152.2228
7 8.749998405666 6765489908816954.4416
8 9.749999880609 42584309749044362.9249
9 10.749997903712 223929011859171157.5936
10 11.750001205877 1015816588633700539.7371
11 12.749999077236 4072305009211161490.4212
12 13.750001119728 14699984721249482813.7254
13 14.749999018489 48487706999631315385.0831
14 15.750000906590 147890502952482182259.1750
15 16.749999084633 421133015211927679457.9063

Dimension 37

n Scaling dimension Δn Degeneracy dn
1 2.764490031484 22530390.9417
2 3.802681049745 5607451366.9995
3 4.810598903942 339332399639.7643
4 5.812305833538 9268115636177.8153
5 6.812519919515 149120875903496.9615
6 7.812522531300 1638623109504139.0497
7 8.812501551411 13485528428057228.5042
8 9.812499658497 88467499675493312.0687
9 10.812498198711 483440550058158836.7540
10 11.812500548498 2273118954027238614.5996
11 12.812499486057 9423879143894923262.6385
12 13.812500707148 35107908235010355187.7237
13 14.812499462746 119298033358084306172.5543
14 15.812500512809 374239027465757182449.9002
15 16.812499474174 1094466013549162377218.9852

Dimension 38

n Scaling dimension Δn Degeneracy dn
1 2.822346174983 32056460.8375
2 3.863625960074 8556469589.1343
3 4.872618640965 550245679995.8393
4 5.874681713216 15890353750999.0674
5 6.875002967660 269166866822682.0876
6 7.875024004607 3101744198589921.0235
7 8.875004800939 26674975188733170.8699
8 9.874999959943 182282487727042855.4366
9 10.874998465787 1034668212377447720.0608
10 11.875000032625 5040609390678117379.9653
11 12.874999716455 21603575678482120310.9555
12 13.875000419426 83036972705412575199.8018
13 14.874999762612 290603579728144773896.7770
14 15.875000261856 937399391438979247186.8969
15 16.874999731726 2814904037094715046240.1617

Dimension 39

n Scaling dimension Δn Degeneracy dn
1 2.880093890529 45549299.4418
2 3.924465325832 13021455118.5186
3 4.934576791385 888790543356.8149
4 5.937029519901 27108789498531.9987
5 6.937475805636 482982853087715.9737
6 7.937523359268 5832017591371909.3336
7 8.937507927752 52377310349458800.5149
8 9.937500732255 372626661914042323.1613
9 10.937498784029 2195981869958519740.6547
10 11.937499650511 11080135166871969626.3603
11 12.937499815498 49076929396900054950.0853
12 13.937500214706 194566316845946962259.6154
13 14.937499953407 701111706078123441566.0065
14 15.937500111623 2324988703992428598527.2826
15 16.937499896193 7167339838516203232556.5720

Dimension 40

n Scaling dimension Δn Degeneracy dn
1 2.937737279944 64638556.9132
2 3.985201196280 19765647157.6728
3 4.996472124315 1430304826002.0725
4 5.999347181456 46027445028465.9983
5 6.999936810975 861746018232733.7294
6 8.000019681310 10895280377517562.7986
7 9.000010631928 102120542718046080.7950
8 10.000001887580 755967589000189683.3853
9 10.999999212393 4623405865475409694.6780
10 11.999999402543 24151658315129730195.5935
11 12.999999826759 110516175411467535833.5392
12 14.000000062685 451789436463209224392.5430
13 15.000000062653 1675853919188206683681.9930
14 16.000000028625 5711930433075498502917.2215
15 16.999999996881 18073063452955291671784.1194