  We prove the rationality of the descendent partition function for stable
pairs on nonsingular toric 3-folds. The method uses a geometric reduction of
the 2- and 3-leg descendent vertices to the 1-leg case. As a consequence, we
prove the rationality of the relative stable pairs partition functions for all
log Calabi-Yau geometries of the form (X,K3) where X is a nonsingular toric
3-fold.
