TAP approach for multi-species spherical spin glasses I: general theory

We develop a generalized TAP approach for the multi-species version of the spherical mixed $p$-spin models. In particular, we prove a generalized TAP representation for the free energy at any overlap vector which is multi-samplable in an appropriate sense. Moreover, we show that if a multi-samplable overlap is maximal, then the TAP correction is equal to an analogue of the well-known Onsager reaction term. Finally, in a companion paper we use the results from the current paper to compute the free energy at any temperature for all multi-species pure $p$-spin models, assuming the free energy converges.