  The Wilsonian renormalization group (RG) requires Euclidean signature. The
conformal factor of the metric then has a wrong-sign kinetic term, which has a
profound effect on its RG properties. Generically for the conformal sector,
complete flows exist only in the reverse direction (i.e. from the infrared to
the ultraviolet). The Gaussian fixed point supports infinite sequences of
composite eigenoperators of increasing infrared relevancy (increasingly
negative mass dimension), which are orthonormal and complete for bare
interactions that are square integrable under the appropriate measure. These
eigenoperators are non-perturbative in $\hbar$ and evanescent. For
$\mathbb{R}^4$ spacetime, each renormalised physical operator exists but only
has support at vanishing field amplitude. In the generic case of infinitely
many non-vanishing couplings, if a complete RG flow exists, it is characterised
in the infrared by a scale $\Lambda_\mathrm{p}>0$, beyond which the field
amplitude is exponentially suppressed. On other spacetimes, of length scale
$L$, the flow ceases to exist once a certain universal measure of inhomogeneity
exceeds $O(1)+2\pi L^2\Lambda^2_\mathrm{p}$. Importantly for cosmology, the
minimum size of the universe is thus tied to the degree of inhomogeneity, with
spacetimes of vanishing size being required to be almost homogeneous. We
initiate a study of this exotic quantum field theory at the interacting level,
and discuss what the full theory of quantum gravity should look like, one which
must thus be perturbatively renormalizable in Newton's constant but
non-perturbative in $\hbar$.
