  A time-reversal invariant topological insulator occupying a Euclidean
half-space determines a 'Quaternionic' self-adjoint Fredholm family. We show
that the discrete spectrum data for such a family is geometrically encoded in a
non-trivial 'Real' gerbe. The gerbe invariant, rather than a na\"ive counting
of Dirac points, precisely captures how edge states completely fill up the bulk
spectral gap in a topologically protected manner.
