  We examine the various indices defined on pairs of almost commuting unitary
matrices that can detect pairs that are far from commuting pairs. We do this in
two symmetry classes, that of general unitary matrices and that of self-dual
matrices, with an emphasis on quantitative results. We determine which values
of the norm of the commutator guarantee that the indices are defined, where
they are equal, and what quantitative results on the distance to a pair with a
different index are possible. We validate a method of computing spin Chern
numbers that was developed with Hastings and only conjectured to be correct.
Specifically, the Pfaffian-Bott index can be computed by the "log method" for
commutator norms up to a specific constant.
