  In four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group
$SU(N)$, we present a closed-form solution for a family of integrated
four-point functions involving stress tensor multiplet composites of arbitrary
R-charge. These integrated correlators are shown to be equivalent to a
one-dimensional semi-infinite lattice of harmonic oscillators with
nearest-neighbor interactions, evolving over the fundamental domain of
$SL(2,\mathbb{Z})$. The solution, exceptionally simple in an
$SL(2,\mathbb{Z})$-invariant eigenbasis, is exact in the R-charge $p$, rank
$N$, and complexified gauge coupling $\tau$. This permits a systematic and
non-perturbative large charge expansion for any $N$ and $\tau$. Especially
novel is a double-scaled "gravity regime" in which $p \sim N^2 \gg 1$,
holographically dual to a large charge regime of semiclassical type IIB string
theory in AdS$_5\, \times$ S$^5$. Our results in this limit provide a
holographic computation of integrated semiclassical string amplitudes at
arbitrary string coupling, including an emergent string scale with a large
charge dressing factor. We compare to extremal correlators in superconformal
QCD, for which we predict new genus expansions at large charge scaling with
$N$.
