  It is very difficult for any nuclear model to pin down the saturation
property and high-density equation of state (EOS) simultaneously because of
high nonlinearity of the nuclear many-body problem. In this work, we propose,
for the first time, to use the special property of light kaonic nuclei to
characterize the relation between saturation property and high-density EOS.
With a series of relativistic mean-field models, this special property is found
to be the level inversion between orbitals $2S_{1/2}$ and $1D_{5/2}$ in light
kaonic nuclei. This level inversion can serve as a theoretical laboratory to
group the incompressibility at saturation density and the EOS at supra-normal
densities simultaneously.
