  It is a long-standing conjecture that any CFT with a large central charge and
a large gap $\Delta_{\text{gap}}$ in the spectrum of higher-spin single-trace
operators must be dual to a local effective field theory in AdS. We prove a
sharp form of this conjecture by deriving numerical bounds on bulk Wilson
coefficients in terms of $\Delta_{\text{gap}}$ using the conformal bootstrap.
Our bounds exhibit the scaling in $\Delta_{\text{gap}}$ expected from
dimensional analysis in the bulk. Our main tools are dispersive sum rules that
provide a dictionary between CFT dispersion relations and S-matrix dispersion
relations in appropriate limits. This dictionary allows us to apply
recently-developed flat-space methods to construct positive CFT functionals. We
show how AdS$_{4}$ naturally resolves the infrared divergences present in 4D
flat-space bounds. Our results imply the validity of twice-subtracted
dispersion relations for any S-matrix arising from the flat-space limit of
AdS/CFT.
