  The asymptotic safety program builds on a high-energy completion of gravity
based on the Reuter fixed point, a non-trivial fixed point of the gravitational
renormalization group flow. At this fixed point the canonical mass-dimension of
coupling constants is balanced by anomalous dimensions induced by quantum
fluctuations such that the theory enjoys quantum scale invariance in the
ultraviolet. The crucial role played by the quantum fluctuations suggests that
the geometry associated with the fixed point exhibits non-manifold like
properties. In this work, we continue the characterization of this geometry
employing the composite operator formalism based on the effective average
action. Explicitly, we give a relation between the anomalous dimensions of
geometric operators on a background $d$-sphere and the stability matrix
encoding the linearized renormalization group flow in the vicinity of the fixed
point. The eigenvalue spectrum of the stability matrix is analyzed in detail
and we identify a "perturbative regime" where the spectral properties are
governed by canonical power counting. Our results recover the feature that
quantum gravity fluctuations turn the (classically marginal) $R^2$-operator
into a relevant one. Moreover, we find strong indications that higher-order
curvature terms present in the two-point function play a crucial role in
guaranteeing the predictive power of the Reuter fixed point.
