  The set of local gauge invariant quantities for linearized gravity on the
Kerr spacetime presented by two of the authors (S.A, T.B.) in
(arXiv:1803.05341) is shown to be complete. In particular, any gauge invariant
quantity for linearized gravity on Kerr that is local and of finite order in
derivatives can be expressed in terms of these gauge invariants and derivatives
thereof. The proof is carried out by constructing a complete compatibility
complex for the Killing operator, and demonstrating the equivalence of the
gauge invariants from (arXiv:1803.05341) with the first compatibility operator
from that complex.
