  Guided by the many-particle quantum theory of interacting systems, we develop
a uniform classification scheme for topological phases of disordered gapped
free fermions, encompassing all symmetry classes of the Tenfold Way. We apply
this scheme to give a mathematically rigorous proof of bulk-boundary
correspondence. To that end, we construct real C$^\ast$-algebras harbouring the
bulk and boundary data of disordered free-fermion ground states. These we
connect by a natural bulk-to-boundary short exact sequence, realising the bulk
system as a quotient of the half-space theory modulo boundary contributions. To
every ground state, we attach two classes in different pictures of real
operator $K$-theory (or $KR$-theory): a bulk class, using Van Daele's picture,
along with a boundary class, using Kasparov's Fredholm picture. We then show
that the connecting map for the bulk-to-boundary sequence maps these
$KR$-theory classes to each other. This implies bulk-boundary correspondence,
in the presence of disorder, for both the "strong" and the "weak" invariants.
