  Using exact optimization methods, we find all of the ground states of +/- h
random-field Ising magnets (RFIM) and of dilute antiferromagnets in a field
(DAFF). The degenerate ground states are usually composed of isolated clusters
(two-level systems) embedded in a frozen background. We calculate the
paramagnetic response (sublattice response) and the ground state entropy for
the RFIM (DAFF) due to these clusters. In both two and three dimensions there
is a broad regime in which these quantities are strictly positive, even at
irrational values of h/J (J is the exchange constant).
