  The binary black hole mergers observed by LIGO-Virgo gravitational-wave
detectors pose two major challenges: (i) how to produce these massive black
holes from stellar processes; and (ii) how to bring them close enough to merge
within the age of the universe? We derive a fundamental constraint relating the
binary separation and the available stellar budget in the universe to produce
the observed black hole mergers. We find that $\lesssim 14\%$ of the entire
budget contributes to the observed merger rate of $(30 + 30)~\mathrm{M}_\odot$
black holes, if the separation is around the diameter of their progenitor
stars. Furthermore, the upgraded LIGO detector and third-generation
gravitational-wave detectors are not expected to find stellar-mass black hole
mergers at high redshifts. From LIGO's strong constraints on the mergers of
black holes in the pair-instability mass-gap ($60-120~\mathrm{M}_\odot$), we
find that $\lesssim 0.8\%$ of all massive stars contribute to a remnant black
hole population in this gap. Our derived separation$-$budget constraint
provides a robust framework for testing the formation scenarios of stellar
binary black holes.
