  We study seven-brane SU(5) GUT models of string phenomenology which can be
consistently analyzed in a purely local framework. The requirement that gravity
can decouple constrains the form of four-dimensional physics as well as the
geometry of spacetime. We rule out a large family of candidate UV completions
of such models and derive a priori constraints on the local singularities of
compact elliptic Calabi-Yau fourfolds. These constraints are strong enough to
obstruct a wide class of brane constructions from UV completion in string
theory. It is demonstrated that consistent local models always have exotic
Yukawa coupling structures, and hidden sectors or interesting non-perturbative
superpotentials which merit further investigation.
