  Celestial amplitudes may be decomposed as weighted integrals of
AdS$_3$-Witten diagrams associated to each leaf of a hyperbolic foliation of
spacetime. We show, for the Kleinian three-point MHV amplitude, that each leaf
subamplitude is smooth except for the expected light-cone singularities.
Moreover, we find that the full translationally-invariant celestial amplitude
is simply the residue of the pole in the leaf amplitude at the point where the
total conformal weights of the gluons equals three. This full celestial
amplitude vanishes up to light-cone contact terms, as required by spacetime
translation invariance, and reduces to the expression previously derived by
Mellin transformation of the Parke-Taylor formula.
