What factor ultimately dictates whether a collapsed core becomes a Neutron Star or a Black Hole?

Answer

The final mass of the remnant core left after the supernova explosion.

While initial mass sets the trajectory, the critical factor determining the final label is the mass of the object left behind after the supernova; exceeding the limit for neutron star stability results in a Black Hole.

What factor ultimately dictates whether a collapsed core becomes a Neutron Star or a Black Hole?
astronomylife cyclestarstellar evolutionmassive stars