  We prove that the degree of the CM line bundle for a normal family over a
curve with fixed general fibers is strictly minimized if the special fiber is
either a smooth projective manifold with a unique cscK metric or ``specially
K-stable", which is a new class we introduce in this paper. This phenomenon, as
conjectured by Odaka (cf., [Oda20]), is a quantitative strengthening of the
separatedness conjecture of moduli spaces of polarized K-stable varieties.
  The above mentioned special K-stability implies the original K-stability and
a lot of cases satisfy it e.g., K-stable log Fano, klt Calabi-Yau (i.e.,
$K_X\equiv0$), lc varieties with the ample canonical divisor and uniformly
adiabatically K-stable klt-trivial fibrations over curves (cf., [Hat22]).
