  As has been known since the 90s, there is an integrable structure underlying
two-dimensional gravity theories. Recently, two-dimensional gravity theories
have regained an enormous amount of attention, but now in relation with quantum
chaos - superficially nothing like integrability. In this paper, we return to
the roots and exploit the integrable structure underlying dilaton gravity
theories to study a late time, large $e^{S_\text{BH}}$ double scaled limit of
the spectral form factor. In this limit, a novel cancellation due to the
integrable structure ensures that at each genus $g$ the spectral form factor
grows like $T^{2g+1}$, and that the sum over genera converges, realising a
perturbative approach to the late-time plateau. Along the way, we clarify
various aspects of this integrable structure. In particular, we explain the
central role played by ribbon graphs, we discuss intersection theory, and we
explain what the relations with dilaton gravity and matrix models are from a
more modern holographic perspective.
