  The second supernova that forms double-neutron-star systems is expected to
occur in a progenitor that is ultra-stripped due to binary interactions. Thus,
the secondary neutron star's mass as well as the post-supernova binary's
orbital parameters will depend on the nature of the collapsing progenitor core.
Since neutron stars are in the strong-gravity regime, their binding energy
makes up a significant fraction of their total mass-energy budget. The second
neutron star's binding energy may thus provide a unique insight as to whether
its progenitor was a low-mass iron core or an oxygen-neon-magnesium core. I
obtain relations for the baryonic mass and binding energy incorporating both a
hadronic equation-of-state catalog as well as recent multi-messenger
neutron-star observations. With these relations, I obtain the first tight
constraints on the baryonic mass and binding energy of three neutron stars that
are thought to have formed from an ultra-stripped progenitor. With these
constraints, I test if each neutron star is consistent with forming from an
ONeMg core that undergoes an electron-capture supernova. From these tests, I
find that this scenario can be ruled out for one of three neutron stars.
Neutron-star properties and the dense-matter equation of state can thus help
distinguish binary formation scenarios.
