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We give a proof of the most general multiple scalar field theory in flatspace-time free of Ostrogradski ghosts. We start from the assumption that theaction is a functional of the scalar fields and their derivatives of order uptotwo (which in general can have field equations of derivative order up to four),and arrive at the most-general field theory that can be constructed such thatthe field equations are of derivative order up to two. Naturally, this theoryincludes all physically viable multi-scalar field theories in flat space-time,in the sense that they don't suffer from Ostrogradski instability \cite{ostro}.

Proof of the most general multiple-scalar field theory in Minkowski space-time free of Ostrogradski Ghost