  We prove a key result regarding the mass and angular momentum content of
linear vacuum perturbations of the Kerr metric obtained through the formalism
developed by Chrzarnowski, Cohen, and Kegeles (CCK). More precisely we prove
that the Abbott-Deser mass and angular momentum integrals of any such
perturbation vanish, when that perturbation was obtained from a regular Fourier
mode of the Hertz potential. As a corollary we obtain a generalization of
previous results on the completion of the `no string' radiation gauge metric
perturbation generated by a point particle. We find that for any bound orbit
around a Kerr black hole, the mass and angular momentum perturbations
completing the CCK metric are simply the energy and angular momentum of the
particle "outside" the orbit and vanish "inside" the orbit.
