  We discuss the holographic description of Narain $U(1)^c\times U(1)^c$
conformal field theories, and their potential similarity to conventional weakly
coupled gravity in the bulk, in the sense that the effective IR bulk
description includes "$U(1)$ gravity" amended with additional light degrees of
freedom. Starting from this picture, we formulate the hypothesis that in the
large central charge limit the density of states of any Narain theory is
bounded by below by the density of states of $U(1)$ gravity. This immediately
implies that the maximal value of the spectral gap for primary fields is
$\Delta_1=c/(2\pi e)$. To test the self-consistency of this proposal, we study
its implications using chiral lattice CFTs and CFTs based on quantum stabilizer
codes. First we notice that the conjecture yields a new bound on quantum
stabilizer codes, which is compatible with previously known bounds in the
literature. We proceed to discuss the variance of the density of states, which
for consistency must be vanishingly small in the large-$c$ limit. We consider
ensembles of code and chiral theories and show that in both cases the density
variance is exponentially small in the central charge.
