  We argue that for any single-trace operator in ${\cal N}=4$ SYM theory there
is a large twist double-scaling limit in which the Feynman graphs have an
iterative structure. Such structure can be recast using a graph-building
operator. Generically, this operator mixes between single-trace operators with
different scaling limits. The mixing captures both the finite coupling spectrum
and corrections away from the large twist limit. We first consider a class of
short operators with gluons and fermions for which such mixing problems do not
arise, and derive their finite coupling spectra. We then focus on a class of
long operators with gluons that do mix. We invert their graph-building operator
and prove its integrability. The picture that emerges from this work opens the
door to a systematic expansion of ${\cal N}=4$ SYM theory around the large
twist limit.
