Ground state energies of multipartite $p$-spin models -- partially lifted RDT view

We consider ground state energies (GSE) of multipartite $p$-spin models. Relying on partially lifted random duality theory (pl RDT) concepts we introduce an analytical mechanism that produces easy to compute lower and upper GSE bounds for \emph{any} spin sets. We uncover that these bounds actually match in case of fully spherical sets thereby providing optimal GSE values for spherical multipartite pure $p$-spin models. Numerical evidence further suggests that our upper and lower bounds may match even in the Ising scenarios. As such developments are rather intriguing, we formulate several questions regarding the connection between our bounds matching generality on the one side and the spin sets structures on the other.