  Motivated by AdS/CFT, we address the following outstanding question in large
$N$ conformal field theory: given the appearance of a single-trace operator in
the ${\cal O}\times{\cal O}$ OPE of a scalar primary ${\cal O}$, what is its
total contribution to the vacuum four-point function $\langle {\cal O}{\cal
O}{\cal O}{\cal O}\rangle$ as dictated by crossing symmetry? We solve this
problem in 4d conformal field theories at leading order in $1/N$. Viewed
holographically, this provides a field theory reconstruction of
crossing-symmetric, four-point exchange amplitudes in AdS$_5$. Our solution
takes the form of a resummation of the large spin solution to the crossing
equations, supplemented by corrections at finite spin, required by crossing.
The method can be applied to the exchange of operators of arbitrary twist
$\tau$ and spin $s$, although it vastly simplifies for even-integer twist,
where we give explicit results. The output is the set of OPE data for the
exchange of all double-trace operators $[{\cal O}{\cal O}]_{n,\ell}$. We find
that the double-trace anomalous dimensions $\gamma_{n,\ell}$ are negative,
monotonic and convex functions of $\ell$, for all $n$ and all $\ell>s$. This
constitutes a holographic signature of bulk causality and classical dynamics of
even-spin fields. We also find that the "derivative relation" between
double-trace anomalous dimensions and OPE coefficients does not hold in
general, and derive the explicit form of the deviation in several cases.
Finally, we study large $n$ limits of $\gamma_{n,\ell}$, relevant for the Regge
and bulk-point regimes.
