What single factor dictates the boundary between forming a neutron star and a black hole?

The remaining mass of the core following the supernova.

The determination between whether a core collapses into a neutron star or continues collapsing into a black hole is almost entirely dependent on the mass remaining in the core after the initial supernova explosion has ejected the outer stellar layers. If this residual mass is below a certain critical threshold, the core settles as a neutron star, supported by internal pressure. However, if the remaining mass is high enough, gravity overcomes all resisting forces, leading to the formation of a black hole. This mass dependence means that the initial mass of the progenitor star is the single most important factor influencing the final compact object left behind.