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We show that all M-theory backgrounds which admit more than 29 Killingspinors are maximally supersymmetric. In particular, we find that thesupercovariant curvature of all backgrounds which preserve 30 supersymmetries,subject to field equations and Bianchi identities, vanishes, and that there areno such solutions which arise as discrete quotients of maximally supersymmetricbackgrounds.

M-theory backgrounds with 30 Killing spinors are maximally supersymmetric