  The free energy of multiple systems of spherical spin glasses with
constrained overlaps was first studied in arXiv:math/0604082. The authors
proved an upper bound of the constrained free energy using Guerra's
interpolation. In this paper, we prove this upper bound is sharp. Our approach
combines the ideas of the Aizenman-Sims-Starr scheme in arXiv:1204.5115 and the
synchronization mechanism used in the vector spin models in arXiv:1512.04441
and arXiv:1512.00370 to prove the matching lower bound. We derive a vector
version of the Aizenman-Sims-Starr scheme for spherical spin glass and use the
synchronization property of arrays obeying the overlap-matrix form of the
Ghirlanda-Guerra identities to prove the matching lower bound.
