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We develop a new approach to extracting the physical consequences ofS-duality of four-dimensional $\mathcal{N}=4$ super Yang-Mills (SYM) and itsstring theory dual, based on $SL(2,\mathbb{Z})$ spectral theory. We observethat CFT observables $\mathcal{O}$, invariant under $SL(2,\mathbb{Z})$transformations of a complexified gauge coupling $\tau$, admit a uniquespectral decomposition into a basis of square-integrable functions. Thisformulation has direct implications for the analytic structure of$\mathcal{N}=4$ SYM data, both perturbatively and non-perturbatively in allparameters. For example, $k$-instanton sectors are uniquely determined by thezero- and one-instanton sectors, and Borel summable series around$k$-instantons have convergence radii with simple $k$-dependence. In large $N$limits, we derive the existence and scaling of non-perturbative effects, whichwe exhibit for certain $\mathcal{N}=4$ SYM observables. An elegant benchmarkfor these techniques is the integrated four-point function conjecturallydetermined by [arXiv:2102.09537] for all $\tau$ for $SU(N)$ gauge group; wederive and elucidate its form.  These results have ramifications for holography. We explain how$\langle\mathcal{O}\rangle$, the ensemble average over the $\mathcal{N}=4$supersymmetric conformal manifold with respect to the Zamolodchikov measure, iscleanly isolated by the spectral decomposition. We prove that the large $N$limit of $\langle\mathcal{O}\rangle$ equals the large $N$, large 't Hooftcoupling limit of $\mathcal{O}$. Holographically speaking,$\langle\mathcal{O}\rangle = \mathcal{O}_{\rm sugra}$, its value in type IIBsupergravity on AdS$_5 \times$ S$^5$. This result, which extends to all ordersin $1/N$, embeds ensemble averaging into the traditional AdS/CFT paradigm. Thestatistics of the $SL(2,\mathbb{Z})$ ensemble exhibit both perturbative andnon-perturbative $1/N$ effects.

Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings