  We present an algorithm for reliably and systematically proving the existence
of spectral gaps in Hamiltonians with quasicrystalline order, based on
numerical calculations on finite domains. We apply this algorithm to prove that
the Hofstadter model on the Ammann-Beenker tiling of the plane has spectral
gaps at certain energies, and we are able to prove the existence of a spectral
gap where previous numerical results were inconclusive. Our algorithm is
applicable to more general systems with finite local complexity and eventually
finds all gaps, circumventing an earlier no-go theorem regarding the
computability of spectral gaps for general Hamiltonians.
