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1461 bytes, as of the version from 2026-10-25 13:49 (current). Recorded here, not run.
"""Check the nonzero-eigenfunction condition for numberdb.org/T488.
Run with SageMath: sage -python check_definition.py
This checks the reported definition omission exactly. It neither computes
nor modifies spectral data. The positive control checks the derivatives.
"""
import json
import numberdb.sage
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.version import version
ring = PolynomialRing(QQ, names=('x', 'y', 'lam', 'nu_x', 'nu_y'))
x, y, lam, nu_x, nu_y = ring.gens()
def laplacian(u):
return u.derivative(x, 2) + u.derivative(y, 2)
def normal_derivative(u):
return nu_x * u.derivative(x) + nu_y * u.derivative(y)
control = x**4 + y**4
assert laplacian(laplacian(control)) == 48
assert normal_derivative(x + 2*y) == nu_x + 2*nu_y
u = ring.zero()
assert laplacian(laplacian(u)) - lam*u == 0
assert u == 0
assert normal_derivative(u) == 0
print(json.dumps({
'sage_version': version,
'control_bilaplacian_x4_plus_y4': '48',
'control_normal_derivative_x_plus_2y': 'nu_x + 2*nu_y',
'zero_solution_pde_residual': '0 for arbitrary lam',
'zero_solution_boundary_value': '0',
'zero_solution_normal_derivative': '0 for arbitrary normal',
'conclusion': 'The zero function satisfies the homogeneous equations for every lam; the definition of an eigenvalue requires a nonzero solution.',
'spectral_values_recomputed': False,
}, indent=2))