  We study correlators of four protected (half-BPS) operators in strongly
coupled supersymmetric Yang-Mills theory. These are dual to tree-level
supergravity amplitudes on AdS${}_5\times$S${}_5$ for various spherical
harmonics on the five-sphere. We use conformal field theory methods, in
particular a recently obtained Lorentzian inversion formula, to analytically
bootstrap these correlators. The extracted $1/N^2$ double-trace anomalous
dimensions confirm a simple pattern recently conjectured by Aprile, Drummond,
Heslop and Paul. We explain this pattern by an unexpected ten-dimensional
conformal symmetry which appears to be enjoyed by tree-level supergravity (or a
suitable subsector of it). The symmetry combines all spherical harmonics into a
single ten-dimensional object, and yields compact expressions for the leading
logarithmic part of any half-BPS correlator at each loop order.
