  We analyze the localization equations relevant to the quantum entropy of
spinning supersymmetric black holes in five-dimensional asymptotically flat
space. The precise problem is to classify all solutions to the off-shell
supersymmetry equations in N=2 supergravity coupled to $n_\text{v}+1$ vector
multiplets around the near-horizon black hole. We rewrite these equations in
terms of the bosonic spinor bilinears that exist in the geometry for an
arbitrary background. We then focus on the vector multiplet fluctuations around
the near-horizon attractor region of the supersymmetric black hole, and
classify all smooth solutions to the localization equations in this background
for different choices of analytic continuation. For the choice of analytic
continuation consistent with the 4d/5d lift, we find that the most general
localization solution for the five-dimensional black hole problem is
an~$(n_\text{v}+1)$-dimensional manifold, which is precisely the lift of the
localization manifold for supersymmetric black holes in four-dimensional
asymptotically flat space.
