We prove that flat Jackiw--Teitelboim (JT) gravity arises as a conformal scaling limit of random flat combinatorial disks. We give a clean criterion, formulated in terms of branching explorations, for a discrete model to converge to this conjecturally universal limit. For simplicity, we apply this criterion to a Boltzmann ensemble of disks with rectilinear boundary, defined by summing over combinatorial types and integrating over lengths. As the length of a distinguished side tends to infinity, the rescaled surface converges in law to a continuum random surface prescribed by Ferrari's path-integral formulation of JT gravity. Convergence is in the topology of smooth convergence of the conformal factors on compact subsets of the interior when uniformizing in a fixed domain; the limiting conformal factor is described by a harmonic Gaussian field closely related to holomorphic multiplicative chaos. The crux is a branching exploration of flat JT gravity whose lengths evolve as conditioned Cauchy processes and whose unexplored components are conditionally independent. We construct this exploration using a conformal welding theorem proved in a companion paper, and show that it is the scaling limit of Budd's row-by-row exploration of the discrete disks, analogous to the peeling of random planar maps. Under a tightness condition, our criterion upgrades this convergence to a conformal scaling limit for the discrete disks. Other natural combinatorial disk models admit analogous first-row recursions, providing a route to their scaling limits.
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