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.
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