  Two recent papers proved that complex index pairings can be calculated as the
half-signature of a finite dimensional matrix, called the spectral localizer.
This paper contains a new proof of this connection for even index pairings
based on a spectral flow argument. It also provides a numerical study of the
spectral gap and the half-signature of the spectral localizer for a typical
two-dimensional disordered topological insulator in the regime of a mobility
gap at the Fermi energy. This regime is not covered by the above mathematical
results (which suppose a bulk gap), but nevertheless the half-signature of the
spectral localizer is a clear indicator of a topological phase.
