  When the $SU(N)$ ${\cal N} = 4$ super-Yang-Mills (SYM) theory with
complexified gauge coupling $\tau$ is placed on a round four-sphere and
deformed by an ${\cal N} = 2$-preserving mass parameter $m$, its free energy
$F(m, \tau, \bar \tau)$ can be computed exactly using supersymmetric
localization. In this work, we derive a new exact relation between the fourth
derivative $\partial_m^4 F(m, \tau, \bar \tau) \big|_{m=0}$ of the sphere free
energy and the integrated stress-tensor multiplet four-point function in the
$\mathcal{N}=4$ SYM theory. We then apply this exact relation, along with
various other constraints derived in previous work (coming from analytic
bootstrap, the mixed derivative $\partial_\tau \partial_{\bar \tau}
\partial_m^2 F(m, \tau, \bar \tau) \big|_{m=0}$, and type IIB superstring
theory scattering amplitudes) to determine various perturbative terms in the
large $N$ and large 't Hooft coupling $\lambda$ expansion of the ${\cal N} = 4$
SYM correlator at separated points. In particular, we determine the leading
large-$\lambda$ term in the ${\cal N} = 4$ SYM correlation function at order
$1/N^8$. This is three orders beyond the planar limit.
