  The unbound nature of pure neutron matter (PNM) requires intrinsic
correlations between the symmetric nuclear matter (SNM) EOS parameters
(incompressibility $K_0$, skewness $J_0$ and kurtosis $I_0$) and those (slope
$L$, curvature $K_{\rm{sym}}$ and skewness $J_{\rm{sym}}$) characterizing the
symmetry energy independent of any nuclear many-body theory. We investigate
these intrinsic correlations and their applications in better constraining the
poorly known high-density behavior of nuclear symmetry energy. Several novel
correlations connecting the characteristics of SNM EOS with those of nuclear
symmetry energy are found. In particular, at the lowest-order of
approximations, the bulk parts of the slope $L$, curvature $K_{\rm{sym}}$ and
skewness $J_{\rm{sym}}$ of the symmetry energy are found to be $L\approx K_0/3,
K_{\rm{sym}}\approx LJ_0/2K_0$ and $J_{\rm{sym}}\approx I_0L/3K_0$,
respectively. High-order corrections to these simple relations can be written
in terms of the small ratios of high-order EOS parameters. The resulting
intrinsic correlations among some of the EOS parameters reproduce very nicely
their relations predicted by various microscopic nuclear many-body theories and
phenomenological models constrained by available data of terrestrial
experiments and astrophysical observations in the literature. The unbound
nature of PNM is fundamental and the required intrinsic correlations among the
EOS parameters characterizing both the SNM EOS and symmetry energy are
universal. These intrinsic correlations provide a novel and model-independent
tool not only for consistency checks but also for investigating the poorly
known high-density properties of neutron-rich matter by using those with
smaller uncertainties.
