  The low energy structures of N=1 supersymmetric models with E_6, F_4 and E_7
gauge groups and fundamental irrep matter contents are studied herein. We
identify sets of gauge invariant composites which label all flat directions in
the confining/Higgs phases of these theories. The impossibility of mapping
several of these primary operators rules out previously conjectured exceptional
self duals reported in the literature.
