High-dynamic-range surface photometry in a companion paper makes possible accurate measurement of the stellar light deficits L _def and mass deficits M _def associated with the cores of elliptical galaxies. We show that L _def correlates with velocity dispersions σ of the host galaxy bulge averaged outside the central region that may be affected by a supermassive black hole (BH). We confirm that L _def correlates with BH mass M _•. Also, the fractional light deficit L _def/L _* correlates with M _•/M _*, the ratio of BH mass to the galaxy stellar mass. All three correlations have scatter similar to or smaller than the scatter in the well-known correlation between M _• and σ. The new correlations are remarkable in view of the dichotomy between ellipticals with cores and those with central extra light. Core light deficit correlates closely with M _• and σ, but extra light does not. This supports the suggestion that extra light Es are made in wet mergers with starbursts whereas core Es are made in dry mergers. After dry mergers, cores are believed to be scoured by BH binaries that fling stars away as their orbits decay or by BHs that sink back to the center after recoiling from anisotropic gravitational radiation emitted when they merge. Direct evidence for these mechanisms has been elusive. We interpret the new correlations as the "smoking gun" that connects cores with BHs. Together, the M _•-σ and M _•-L _def correlations give us two independent ways to estimate BH masses in core Es.