  We present a numerical analysis supporting the evidence that the redshift
evolution of the drifting coefficient of the field cluster mass function is
capable of breaking several cosmic degeneracies. This evidence is based on the
data from the CoDECS and DUSTGRAIN-pathfinder simulations performed separately
for various non-standard cosmologies including coupled dark energy, $f(R)$
gravity and combinations of $f(R)$ gravity with massive neutrinos as well as
for the standard $\Lambda$CDM cosmology. We first numerically determine the
field cluster mass functions at various redshifts in the range of $0\le z\le 1$
for each cosmology. Then, we compare the analytic formula developed in previous
works with the numerically obtained field cluster mass functions by adjusting
its drifting coefficient, $\beta$, at each redshift. It is found that the
analytic formula with the best-fit coefficient provides a good match to the
numerical results at all redshifts for all of the cosmologies. The empirically
determined redshift evolution of the drifting coefficient, $\beta(z)$, turns
out to significantly differ among different cosmologies. It is also shown that
even without using any prior information on the background cosmology the
drifting coefficient, $\beta(z)$, can discriminate with high statistical
significance the degenerate non-standard cosmologies not only from the
$\Lambda$CDM but also from one another. It is concluded that the evolution of
the departure from the Einstein-de Sitter state and spherically symmetric
collapse processes quantified by $\beta(z)$ is a powerful probe of gravity and
dark sector physics.
