  The spectrum of IIB supergravity on AdS${}_5 \times S^5$ contains a number of
bound states described by long double-trace multiplets in $\mathcal{N}=4$ super
Yang-Mills theory at large 't Hooft coupling. At large $N$ these states are
degenerate and to obtain their anomalous dimensions as expansions in
$\tfrac{1}{N^2}$ one has to solve a mixing problem. We conjecture a formula for
the leading anomalous dimensions of all long double-trace operators which
exhibits a large residual degeneracy whose structure we describe. Our formula
can be related to conformal Casimir operators which arise in the structure of
leading discontinuities of supergravity loop corrections to four-point
correlators of half-BPS operators.
