Universal identities for Euclidean mechanical Lie brackets

We prove that Euclidean kinetic energy and arbitrary smooth potentials, considered in all dimensions, satisfy no universal Lie identities beyond those of the free polynomially graded Lie algebra of classical mechanics. Polynomial potentials already suffice to separate every nonzero element. The proof reduces positive momentum degree to degree zero: a sufficiently large binary-tree potential distinguishes words in degree one, while in higher degree a free Lie projection and a retraction recover a nonzero multiple of the original element. Symmetrization in a free pre-Lie algebra and finite Taylor jets then separate the potential part. This answers the Euclidean mechanical conjecture of McLachlan and Murua. Exact computations through order eighteen provide supporting finite verification.