  We study the Thouless-Anderson-Palmer (TAP) equations for spin glasses on the
hypercube. First, using a random, approximately ultrametric decomposition of
the hypercube, we decompose the Gibbs measure, $\langle\cdot\rangle_N$, into a
mixture of conditional laws, $\langle\cdot\rangle_{\alpha,N}$. We show that the
TAP equations hold for the spin at any site with respect to
$\langle\cdot\rangle_{\alpha,N}$ simultaneously for all $\alpha$. This result
holds for generic models provided that the Parisi measure of the model has a
jump at the top of its support.
