E-010 — Zeros of Suzuki W(a,theta;z) via Galerkin deficiency vectors; EXTENDED sweep (a<=8, lambda=-1 added, h_conv to 2.5e-3)
08-17-2026 · 48 rows · script
research/numerics/scripts/E-010-suzuki-W-function.py
Metadata
deficiency vectors v_pm from Galerkin resolvent (E-009b machinery, bordered mean-zero solve); W(a,theta;z) per eq. 1.11; real-gauge zero finding on (0,60]; comparison with zeta zero ordinates
Validation checks
| v_minus_symmetry | PASS |
| T_norms_agree | PASS |
| theta0_real_gauge | PASS |
| h_convergence_first_zero | PASS |
Key values
| gamma_1..6 | [14.134725141734695, 21.022039638771556, 25.01085758014569, 30.424876125859512, 32.93506158773919, 37.586178158825675] |
| a=2.0 | [1.5781071479559565, 3.156279708637713, 4.734589954016487, 6.313126214353641, 7.892008843882063, 9.471425811293946, 11.051745119397912, 12.634215895261782, 14.150589728489816, 15.79509494269096, 17.38701469967809, 18.99530256627463] |
| a=4.0 | [0.78646467037824, 1.5729312318082147, 2.3594016186812166, 3.1458778549196453, 3.9323621057289406, 4.7188567403698025, 5.505364405687678, 6.291888130339955, 7.078431467455094, 7.864998671463607, 8.651595018454742, 9.438227296565794] |
| a=6.0 | [0.5239397465249559, 1.0478796489834181, 1.571819864560504, 2.0957605529866874, 2.6197018778056727, 3.143644007875613, 3.667587119190764, 4.191531396047481, 4.715477034091082, 5.23942423978668, 5.7633732415498, 6.287324277308421] |
| a=8.0 | [0.3928980474237194, 0.7857961991893093, 1.1786945600926104, 1.5715932361283171, 1.9644923351809158, 2.357391967431892, 2.7502922464497574, 3.143193289370707, 3.5360952181834935, 3.92899815947609, 4.321902247297599, 4.714807622114] |
| h_conv_first_zero | [0.7867814646177885, 0.78646467037824, 0.7862960897194806] |
| lambda_sensitivity | {'-0.5': [0.786549044345905, 1.573100135109588, 2.359655366162599, 3.14621692956192], '-1.0': [0.78646467037824, 1.5729312318082147, 2.3594016186812166, 3.1458778549196453], '-2.0': [0.786360915811139, 1.5727234886518329, 2.359089412936245, 3.1454604602135774]} |
| theta_sensitivity | {'0.0': [0.78646467037824, 1.5729312318082147, 2.3594016186812166, 3.1458778549196453, 3.9323621057289406, 4.7188567403698025], '1.5707963267948966': [0.9831559835641227, 1.785143661842912, 2.5782009073997156, 3.368499578391118, 4.157616557181577, 4.94614687241914], '3.141592653589793': [1.1256469139766037, 1.935274990583408, 2.731660080425844, 3.5238773590800987, 4.314268719282322, 5.103729551791709]} |
Parameters
| a_sweep | [2.0, 4.0, 6.0, 8.0] |
| h_sweep | 0.005 |
| h_conv | [0.01, 0.005, 0.0025] |
| lambdas | [-0.5, -1.0, -2.0] |
| thetas | [0.0, 1.5707963267948966, 3.141592653589793] |
| z_max | 60.0 |
Caveats
- theta=0 extension has no known zeta-alignment guarantee (Q-S3: no distinguished theta in the paper)
- small a (<=8) — Cor 1.6 concerns a -> infinity; zeros need not be near gamma_j yet; trend with a is the observable
Chart
Data preview (48 rows total)
| a | h | lambda | theta | j | z_j | gamma_j | z_over_gamma |
|---|---|---|---|---|---|---|---|
| 2.0 | 0.005 | -1.0 | 0.0 | 1 | 1.5781071480 | 14.1347251417 | 0.111648 |
| 2.0 | 0.005 | -1.0 | 0.0 | 2 | 3.1562797086 | 21.0220396388 | 0.150141 |
| 2.0 | 0.005 | -1.0 | 0.0 | 3 | 4.7345899540 | 25.0108575801 | 0.189301 |
| 2.0 | 0.005 | -1.0 | 0.0 | 4 | 6.3131262144 | 30.4248761259 | 0.207499 |
| 2.0 | 0.005 | -1.0 | 0.0 | 5 | 7.8920088439 | 32.9350615877 | 0.239623 |
| 2.0 | 0.005 | -1.0 | 0.0 | 6 | 9.4714258113 | 37.5861781588 | 0.251992 |
| 2.0 | 0.005 | -1.0 | 0.0 | 7 | 11.0517451194 | 40.9187190121 | 0.270090 |
| 2.0 | 0.005 | -1.0 | 0.0 | 8 | 12.6342158953 | 43.3270732809 | 0.291601 |
| 2.0 | 0.005 | -1.0 | 0.0 | 9 | 14.1505897285 | 48.0051508812 | 0.294772 |
| 2.0 | 0.005 | -1.0 | 0.0 | 10 | 15.7950949427 | 49.7738324777 | 0.317337 |
| 2.0 | 0.005 | -1.0 | 0.0 | 11 | 17.3870146997 | 52.9703214777 | 0.328241 |
| 2.0 | 0.005 | -1.0 | 0.0 | 12 | 18.9953025663 | 56.4462476971 | 0.336520 |
| 4.0 | 0.005 | -1.0 | 0.0 | 1 | 0.7864646704 | 14.1347251417 | 0.055641 |
| 4.0 | 0.005 | -1.0 | 0.0 | 2 | 1.5729312318 | 21.0220396388 | 0.074823 |
| 4.0 | 0.005 | -1.0 | 0.0 | 3 | 2.3594016187 | 25.0108575801 | 0.094335 |
| 4.0 | 0.005 | -1.0 | 0.0 | 4 | 3.1458778549 | 30.4248761259 | 0.103398 |
| 4.0 | 0.005 | -1.0 | 0.0 | 5 | 3.9323621057 | 32.9350615877 | 0.119397 |
| 4.0 | 0.005 | -1.0 | 0.0 | 6 | 4.7188567404 | 37.5861781588 | 0.125548 |
| 4.0 | 0.005 | -1.0 | 0.0 | 7 | 5.5053644057 | 40.9187190121 | 0.134544 |
| 4.0 | 0.005 | -1.0 | 0.0 | 8 | 6.2918881303 | 43.3270732809 | 0.145218 |
| 4.0 | 0.005 | -1.0 | 0.0 | 9 | 7.0784314675 | 48.0051508812 | 0.147451 |
| 4.0 | 0.005 | -1.0 | 0.0 | 10 | 7.8649986715 | 49.7738324777 | 0.158015 |
| 4.0 | 0.005 | -1.0 | 0.0 | 11 | 8.6515950185 | 52.9703214777 | 0.163329 |
| 4.0 | 0.005 | -1.0 | 0.0 | 12 | 9.4382272966 | 56.4462476971 | 0.167207 |
| 6.0 | 0.005 | -1.0 | 0.0 | 1 | 0.5239397465 | 14.1347251417 | 0.037068 |