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Abstract When the SU(N) N $$ \mathcal{N} $$ = 4 super-Yang-Mills (SYM) theory with complexified gauge coupling τ is placed on a round four-sphere and deformed by an N $$ \mathcal{N} $$ = 2-preserving mass parameter m, its free energy F (m, τ, τ ¯ $$ \overline{\tau} $$ ) can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative ∂ m 4 F m τ τ ¯ m = 0 $$ {\partial}_m^4F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. $$ of the sphere free energy and the integrated stress-tensor multiplet four-point function in the N $$ \mathcal{N} $$ = 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 ∂ τ ∂ τ ¯ ∂ m 2 F m τ τ ¯ m = 0 $$ {\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. $$ , and type IIB superstring theory scattering amplitudes) to determine various perturbative terms in the large N and large ’t Hooft coupling λ expansion of the N $$ \mathcal{N} $$ = 4 SYM correlator at separated points. In particular, we determine the leading large-λ term in the N $$ \mathcal{N} $$ = 4 SYM correlation function at order 1/N 8. This is three orders beyond the planar limit.