Binary Dynamics through the Fifth Power of Spin at O(G^{2}).

We use a previously developed scattering-amplitudes-based framework for determining two-body Hamiltonians for generic binary systems with arbitrary spin S. By construction this formalism bypasses difficulties with unphysical singularities or higher-time derivatives. This framework has been previously used to obtain the exact velocity dependence of the O(G^{2}) quadratic-in-spin two-body Hamiltonian.