  This paper concerns the $\mathbb{Z}_2$ classification of Fermionic
Time-Reversal (FTR) symmetric partial differential Hamiltonians on the
Euclidean plane. We consider the setting of two insulators separated by an
interface. Hamiltonians that are invariant with respect to spatial translations
along the interface are classified into two categories depending on whether
they may or may not be gapped by continuous deformations. Introducing a related
odd-symmetric Fredholm operator, we show that the classification is stable
against FTR-symmetric perturbations.
  The property that non-trivial Hamiltonians cannot be gapped may be
interpreted as a topological obstruction to Anderson localization: no matter
how much (spatially compactly supported) perturbations are present in the
system, a certain amount of transmission in both directions is guaranteed in
the nontrivial phase. We present a scattering theory for such systems and show
numerically that transmission is indeed guaranteed in the presence of
FTR-symmetric perturbations while it no longer is for non-symmetric
fluctuations.
