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The deleterious low-temperature tetragonal phases in prototypical Pt-based superalloys have variously been reported as taking the tI16-U₃Si (DO_c), tI16-Ir₃Si (DO_c′) and tP16-Pt₃Ga structure-types in contrast to the high-temperature cubic cP4-Cu₃Au (L1₂) phase. We have investigated the relative stability of these four structure-types at absolute zero by using density functional theory. We find that the ground state of stoichiometric Pt₃Al is tP16-Pt₃Ga and that the other three lattices are mechanically unstable at absolute zero. Experiments are needed to measure the internal displacement parameters of these three competing tetragonal phases.