  In [Physical Magazine, 35(3):593-601, 1977], Thouless, Anderson, and Palmer
derived a representation for the free energy of the Sherrington-Kirkpatrick
model, called the TAP free energy, written as the difference of the energy and
entropy on the extended configuration space of local magnetizations with an
Onsager correction term. In the setting of mixed $p$-spin models with Ising
spins, we prove that the free energy can indeed be written as the supremum of
the TAP free energy over the space of local magnetizations whose
Edwards-Anderson order parameter (self-overlap) is to the right of the support
of the Parisi measure. Furthermore, for generic mixed $p$-spin models, we prove
that the free energy is equal to the TAP free energy evaluated on the local
magnetization of any pure state.
