The highly frustrated Heisenberg antiferromagnet on Checkerboard and Pyrochlore lattices is subject to strong quantum fluctuations. This problem is amenable to the Contractor Renormalization (CORE) algorithm, which systematically computes the effective interactions in a basis of local cluster eigenstates. CORE was recently used to derive the Plaquette Boson-Fermion model for the square lattice Hubbard model (cond-mat/0108087). For the spin half Checkerboard and Pyrochlore cases, we find singlet ground states which break lattice symmetry, and a gap to spin one excitations. The low excitations are singlets, coupled by Ising like interactions at an emergent low energy scale, which for the Pyrochlore is about 1% of the Heisenberg exchange. We discuss low temperature thermodynamics and new interpretations of finite size numerical data. We argue that our results are common to many models of quantum frustration.