Hofstadter Monarch

The Harper equation psi(n+1) + psi(n-1) + 2 cos(2 pi alpha n + nu) psi(n) = E psi(n) solved at every rational flux alpha = p / 811, with ten Bloch samples per flux and about 6.6 million eigenvalues accumulated as a density of states. The spectrum depends on the two Bloch phases only through cos theta1 + cos theta2, verified to 3.8e-15, which keeps every matrix real symmetric. Energy runs across the frame and flux up it, and the threads brighten at each band edge, a van Hove singularity.