  Given an Equation of State (EOS) for neutron star (NS) matter, there is a
unique mass-radius sequence characterized by a maximum mass
$M_{\rm{NS}}^{\max}$ at radius $R_{\max}$. We first show analytically that the
$M_{\rm{NS}}^{\max}$ and $R_{\max}$ scale linearly with two different
combinations of NS central pressure $P_{\rm{c}}$ and energy density
$\varepsilon_{\rm{c}}$ by dissecting perturbatively the dimensionless
Tolman-Oppenheimer-Volkoff (TOV) equations governing NS internal variables. The
scaling relations are then verified via 87 widely used and rather diverse
phenomenological as well as 17 microscopic NS EOSs with/without considering
hadron-quark phase transitions and hyperons by solving numerically the original
TOV equations. The EOS of densest NS matter allowed before it collapses into a
black hole (BH) is then obtained. Using the universal $M_{\rm{NS}}^{\max}$ and
$R_{\max}$ scalings and NICER (Neutron Star Interior Composition Explorer) and
XMM-Newton mass-radius observational data for PSR J0740+6620, a very narrow
constraining band on the NS central EOS is extracted directly from the data for
the first time without using any specific input EOS model.
