  Conformal theory correlators are characterized by the spectrum and three-
point functions of local operators. We present a formula which extracts this
data as an analytic function of spin. In analogy with a classic formula due to
Froissart and Gribov, it is sensitive only to an "imaginary part" which appears
after analytic continuation to Lorentzian signature, and it converges thanks to
recent bounds on the high-energy Regge limit. At large spin, substituting in
cross-channel data, the formula yields 1/J expansions with controlled errors.
In large-N theories, the imaginary part is saturated by single-trace operators.
For a sparse spectrum, it manifests the suppression of bulk higher-derivative
interactions that constitutes the signature of a local gravity dual in
Anti-de-Sitter space.
