Thouless-Anderson-Palmer equations for the generic p-spin glass model

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, $⟨\cdot {⟩}_N$, into a mixture of conditional laws, $⟨\cdot {⟩}_{\alpha ,N}$. We show that the TAP equations hold for the spin at any site with respect to $⟨\cdot {⟩}_{\alpha ,N}$ 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.