  We construct a fully equivariant correspondence between Gromov-Witten and
stable pairs descendent theories for toric 3-folds X. Our method uses geometric
constraints on descendents, A_n surfaces, and the topological vertex. The
rationality of the stable pairs descendent theory plays a crucial role in the
definition of the correspondence. We prove our correspondence has a
non-equivariant limit.
  As a result of the construction, we prove an explicit non-equivariant
stationary descendent correspondence for X (conjectured previously by MNOP).
Using descendent methods, we establish the relative GW/Pairs correspondence for
X/D in several basic new log Calabi-Yau geometries. Among the consequences is a
rationality constraint for non-equivariant descendent Gromov-Witten series for
P^3.
