  The gravitational asymptotic safety program envisions the high-energy
completion of gravity through an interacting renormalization group fixed point,
the Reuter fixed point. The predictive power of the construction is encoded in
the spectrum of the stability matrix which is obtained from linearizing the
renormalization group flow around this fixed point. A key result of the
asymptotic safety program is that parts of this spectrum exhibits an
almost-Gaussian scaling behavior, entailing that operators which are
classically highly UV-irrelevant do not induce new free parameters. In this
article, we track down the origin of this property by contrasting the structure
of the stability matrix computed from the Wetterich equation and the Composite
Operator equation within the realm of f(R)-truncations. We show that the
almost-Gaussian scaling is not linked to the classical part of the beta
functions. It is a quantum-induced almost-Gaussian scaling originating from the
quantum corrections in the flow equation. It relies on a subtle interplay among
the analytic structure of the theory's two-point function and the way the
Wetterich equation integrates out fluctuation modes. As a byproduct we
determine the parts of the eigenmode spectrum which is robust with respect to
changing the regularization procedure.
