  In a companion paper we developed the generalized TAP approach for general
multi-species spherical mixed $p$-spin models. In this paper, we use it to
compute the limit of the free energy at any temperature for all pure
multi-species spherical $p$-spin models, assuming that certain free energies
converge. Importantly, the pure multi-species models do not satisfy the
convexity assumption on the mixture which was crucial in the recent proofs of
the Parisi formula for the multi-species Sherrington-Kirkpatrick model by Barra
et al. (2015) and Panchenko (2015) and for the multi-species spherical mixed
p-spin models by Bates and Sohn (2021).
