E-009b — Galerkin spectrum of Suzuki A_a (Weil-form operator); validated vs zero-side explicit formula

08-17-2026 · 120 rows · script research/numerics/scripts/E-009-suzuki-galerkin.py

Metadata

P1-hat Galerkin discretization of the Weil-form operator A_a (Friedrichs ext of D*G_aD, mean-zero constraint); Toeplitz stiffness via exact piecewise Gauss-Legendre quadrature of g; bottom 12 eigenvalues; absolute validation vs zero-side explicit formula

Validation checks

s_values_vs_mpmath	PASS
weil_form_value_matches_zero_side	PASS
eigenvalues_ge_minus_noise_floor	PASS
h_convergence_gaps_shrink	PASS
lambda1_nonincreasing_in_a	PASS

Key values

min_eigenvalue	`0.0000000000005070939098005517`
weil_validation	`{'vTQv': 0.0002437005343030248, 'zero_side': 0.000243701628372111}`
h_conv_lambda1	`{'0.01': 8.834153655139757e-11, '0.005': 1.141180807029745e-11, '0.002': 5.070939098005517e-13}`
lambda1_vs_a	`{'0.25': 0.41385415106618084, '0.5': 0.00020695852905775012, '1.0': 1.4413606342760227e-09, '2.0': 7.43595759028353e-11, '3.0': 1.141180807029745e-11, '4.0': 5.6202942124986806e-12, '5.0': 1.6846008247033063e-12}`
lambda2_vs_a	`{'0.25': 0.8806210575518238, '0.5': 0.01262576697324307, '1.0': 4.71713570880733e-09, '2.0': 2.1537876457934212e-10, '3.0': 4.183030215443876e-11, '4.0': 9.494462112107129e-12, '5.0': 5.729964509284219e-12}`

Parameters

a_sweep	`[0.25, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0]`
h_sweep	`0.005`
h_convergence_at_a	`3.0`
h_conv	`[0.01, 0.005, 0.002]`
n_eig	`12`
gl_points	`12`

Caveats

bottom eigenvalue lambda_a is POSITIVE but very small (~1e-6 scale at a<=5) — earlier table-quadrature version showed spurious -1e-4 values from 1e-3-mesh interpolation noise amplified by 1/h^2; fixed by exact quadrature
eigenvalues of A_a are NOT the zero ordinates: the zeta-aligned operator is Dbar_{a,theta} (first-order, in the T_a-inner-product space) — its spectrum = zeros of W(a,theta;z) (eq 1.11), a separate computation (E-010 candidate)
mean-zero constraint enforced (theorem's operator is on L^2_0)

Chart

Data preview (120 rows total)

a	h	N	j	lambda_j	gamma_j	lambda_over_gamma2
3.0	0.01	600	1	8.8341536551e-11	14.1347251417	4.421710e-13
3.0	0.01	600	2	3.4305776913e-10	21.0220396388	7.762785e-13
3.0	0.01	600	3	2.8457268373e-09	25.0108575801	4.549211e-12
3.0	0.01	600	4	2.9076621594e-09	30.4248761259	3.141133e-12
3.0	0.01	600	5	5.1728425615e-09	32.9350615877	4.768835e-12
3.0	0.01	600	6	1.8364578842e-08	37.5861781588	1.299944e-11
3.0	0.01	600	7	1.9516852046e-08	40.9187190121	1.165643e-11
3.0	0.01	600	8	2.7019811024e-08	43.3270732809	1.439341e-11
3.0	0.01	600	9	5.9424692535e-08	48.0051508812	2.578643e-11
3.0	0.01	600	10	8.2294095248e-08	49.7738324777	3.321747e-11
3.0	0.01	600	11	1.2519278693e-07	52.9703214777	4.461841e-11
3.0	0.01	600	12	1.5658535698e-07	56.4462476971	4.914519e-11
3.0	0.005	1200	1	1.1411808070e-11	14.1347251417	5.711889e-14
3.0	0.005	1200	2	4.1830302154e-11	21.0220396388	9.465451e-14
3.0	0.005	1200	3	1.5381578784e-10	25.0108575801	2.458916e-13
3.0	0.005	1200	4	2.2306921279e-10	30.4248761259	2.409806e-13
3.0	0.005	1200	5	5.1818144785e-10	32.9350615877	4.777106e-13
3.0	0.005	1200	6	5.9518864837e-10	37.5861781588	4.213066e-13
3.0	0.005	1200	7	1.3235087349e-09	40.9187190121	7.904652e-13
3.0	0.005	1200	8	3.0799956551e-09	43.3270732809	1.640708e-12
3.0	0.005	1200	9	4.3404966283e-09	48.0051508812	1.883492e-12
3.0	0.005	1200	10	6.4400545580e-09	49.7738324777	2.599485e-12
3.0	0.005	1200	11	6.5381966032e-09	52.9703214777	2.330198e-12
3.0	0.005	1200	12	1.1868524182e-08	56.4462476971	3.725002e-12
3.0	0.002	3000	1	5.0709390980e-13	14.1347251417	2.538129e-15