  We examine the double-trace spectrum of $\mathcal{N} = 4$ super Yang-Mills
theory in the supergravity limit. At large $N$ double-trace operators exhibit
degeneracy. By considering free-field and tree-level supergravity contributions
to four-point functions of half-BPS operators we resolve the degeneracy for a
large family of double-trace operators. The mixing problem reveals a
surprisingly simple structure which allows us to obtain their three-point
functions at leading order in the large $N$ expansion as well as their leading
anomalous dimensions.
