Theorem: omega(k = pi/a)^2 - 4K/m = 0 (Born-von Karman Brillouin-zone-edge max frequency)

Theorem (Born-vK-zone-edge canonical): for the 1D monatomic-chain dispersion omega^2 = (4K/m) sin^2(ka/2), the Brillouin-zone-edge wavevector k = pi/a gives sin(ka/2) = sin(pi/2) = 1, so omega_max^2 = 4K/m exactly. Canonical sympy pins:…