E-016 — Herglotz fine structure of W at theta=pi vs F (Q-S3c)
08-18-2026 · 3519 rows · script
research/numerics/scripts/E-016-herglotz-fine-structure.py
Metadata
Poisson-smoothed zero density -(1/pi) Im W'/W(iy) at theta=pi (and theta=0 control) vs rho_F; analytic Galerkin z-derivative; zero-Poisson reconstruction cross-check
Validation checks
| V1_plateau_density | PASS |
| V2_rhoF_finitediff | PASS |
| V3_poisson_reconstruction | FAIL |
| V4_vmirror_symmetry | PASS |
Key values
| xi_flint_crosscheck_rel | 0.00000000000000008821833278620677 |
| rhoF_finitediff_max_rel | 0.000000013322584092623682 |
| metrics | {'4.0': {'n_zeros': 381, 'z1': 1.1260734006983593, 'plateau': 1.2713747663068067, 'density_a_over_pi': 1.2732395447351628, 'plateau_rel_dev': 0.0014645935527740487, 'v3_rel': 0.32451368257591473, 'kappa_shift': -0.21982980149584136, 'd_pi': 1.2200734562385358, 'd_zero': 1.2165204320831793}, '6.0': {'n_zeros': 572, 'z1': 0.7499016225074712, 'plateau': 1.9083309455490012, 'density_a_over_pi': 1.909859317102744, 'plateau_rel_dev': 0.00080025347419903, 'v3_rel': 0.2938167612479611, 'kappa_shift': -0.39086861455767985, 'd_pi': 1.855357135046592, 'd_zero': 1.8529753375094085}, '8.0': {'n_zeros': 763, 'z1': 0.5621012628096893, 'plateau': 2.5451204974114527, 'density_a_over_pi': 2.5464790894703255, 'plateau_rel_dev': 0.0005335178539225225, 'v3_rel': 0.2792472337188209, 'kappa_shift': -0.5618837715660804, 'd_pi': 2.4914430957151077, 'd_zero': 2.489482228965594}, '10.0': {'n_zeros': 954, 'z1': 0.44959700551214055, 'plateau': 3.1817851862991344, 'density_a_over_pi': 3.183098861837907, 'plateau_rel_dev': 0.00041270334218086014, 'v3_rel': 0.27086948306792724, 'kappa_shift': -0.7328752211905654, 'd_pi': 3.127620084707738, 'd_zero': 3.126136732274815}} |
| vmirror_sym_err | 0.00000000000003797860165317667 |
Parameters
| a_sweep | [4.0, 6.0, 8.0, 10.0] |
| h | 0.01 |
| lambda | -1.0 |
| y_grid | [0.5, 20.0, 391] |
| z_max_zeros | 300.0 |
Caveats
- Galerkin discretization h=0.01 (E-010 h-convergence: z1 rel drift 4e-4 between h=0.01 and 0.005 at a=4)
- Poisson smoothing at height y averages over scale ~y: lattice oscillations at spacing pi/a are suppressed ~e^{-2ay}; the sensitive region is y <~ few (low zeros resolve individually)
- theta=pi vs theta=0 comparison is the sharp test of the H4 read-off (D-023): if theta=pi is correct, d_pi should lie systematically below d_zero as a grows
Chart
Data preview (3519 rows total)
| series | a | y | rho |
|---|---|---|---|
| F | - | 0.500000 | 6.3713110109e-01 |
| F | - | 0.550000 | 5.7930752798e-01 |
| F | - | 0.600000 | 5.3112965737e-01 |
| F | - | 0.650000 | 4.9037154136e-01 |
| F | - | 0.700000 | 4.5544321587e-01 |
| F | - | 0.750000 | 4.2517870604e-01 |
| F | - | 0.800000 | 3.9870352957e-01 |
| F | - | 0.850000 | 3.7534896368e-01 |
| F | - | 0.900000 | 3.5459488962e-01 |
| F | - | 0.950000 | 3.3603068644e-01 |
| F | - | 1.000000 | 3.1932785647e-01 |
| F | - | 1.050000 | 3.0422047221e-01 |
| F | - | 1.100000 | 2.9049095584e-01 |
| F | - | 1.150000 | 2.7795956840e-01 |
| F | - | 1.200000 | 2.6647652675e-01 |
| F | - | 1.250000 | 2.5591601235e-01 |
| F | - | 1.300000 | 2.4617156274e-01 |
| F | - | 1.350000 | 2.3715248708e-01 |
| F | - | 1.400000 | 2.2878104978e-01 |
| F | - | 1.450000 | 2.2099023685e-01 |
| F | - | 1.500000 | 2.1372196898e-01 |
| F | - | 1.550000 | 2.0692566047e-01 |
| F | - | 1.600000 | 2.0055704835e-01 |
| F | - | 1.650000 | 1.9457723439e-01 |
| F | - | 1.700000 | 1.8895189622e-01 |