  Large scale calculations are performed to establish the global mass
dependence of the nuclear symmetry energy, $a_{sym}(A)$, which in turn depends
on two basic ingredients: the mean-level spacing, $\epsilon(A)$, and the
effective strength of the isovector mean-potential, $\kappa(A)$. Surprisingly,
our results reveal that in modern parameterizations including SLy4, SkO, SkXc,
and SkP these two basic ingredients of $a_{sym}$ are almost equal after
rescaling them linearly by the isoscalar and the isovector effective masses,
respectively. This result points toward a new fundamental property of the
nuclear interaction that remains to be resolved. In addition, our analysis
determines the ratio of the surface-to-volume contributions to $a_{sym}$ to be
$\sim$1.6, consistent with hydrodynamical estimates for the static dipole
polarizability as well as the neutron-skin.
