  The number of local operators in a CFT below a given twist grows with spin.
Consistency with analyticity in spin then requires that at low spin, infinitely
many Regge trajectories must decouple from local correlation functions,
implying infinitely many vanishing conditions for OPE coefficients. In this
paper we explain the mechanism behind this infinity of zeros. Specifically, the
mechanism is related to the two-point function rather than the three-point
function, explaining the vanishing of OPE coefficients in every correlator from
a single condition. We illustrate our result by studying twist-4 Regge
trajectories in the Wilson--Fisher CFT at one loop.
