  In single-metric approximations to the exact renormalization group (RG) for
quantum gravity, it has been not been clear how to treat the large curvature
domain beyond the point where the effective cutoff scale $k$ is less than the
lowest eigenvalue of the appropriate modified Laplacian. We explain why this
puzzle arises from background dependence, resulting in Wilsonian RG concepts
being inapplicable. We show that when properly formulated over an ensemble of
backgrounds, the Wilsonian RG can be restored. This in turn implies that
solutions should be smooth and well defined no matter how large the curvature
is taken. Even for the standard single-metric type approximation schemes, this
construction can be rigorously derived by imposing a modified Ward identity
(mWI) corresponding to rescaling the background metric by a constant factor.
However compatibility in this approximation requires the space-time dimension
to be six. Solving the mWI and flow equation simultaneously, new variables are
then derived that are independent of overall background scale.
