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 in 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.