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Given an Equation of State (EOS) for neutron star (NS) matter, there is aunique 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 differentcombinations of NS central pressure $P_{\rm{c}}$ and energy density$\varepsilon_{\rm{c}}$ by dissecting perturbatively the dimensionlessTolman-Oppenheimer-Volkoff (TOV) equations governing NS internal variables. Thescaling relations are then verified via 87 widely used and rather diversephenomenological as well as 17 microscopic NS EOSs with/without consideringhadron-quark phase transitions and hyperons by solving numerically the originalTOV equations. The EOS of densest NS matter allowed before it collapses into ablack hole (BH) is then obtained. Using the universal $M_{\rm{NS}}^{\max}$ and$R_{\max}$ scalings and NICER (Neutron Star Interior Composition Explorer) andXMM-Newton mass-radius observational data for PSR J0740+6620, a very narrowconstraining band on the NS central EOS is extracted directly from the data forthe first time without using any specific input EOS model.

Core States of Neutron Stars from Anatomizing their Scaled Structure Equations