  We introduce the Merger Entropy Index ($\mathcal{I}_\mathrm{BBH}$), a new
parameter to measure the efficiency of entropy transfer for any generic binary
black hole merger in General Relativity. We find that
$\mathcal{I}_\mathrm{BBH}$ is bounded between an asymptotic maximum and
minimum. For the observed population of mergers detected by LIGO and Virgo, we
find that $\mathcal{I}_\mathrm{BBH}$ is $\lesssim30\%$ of its theoretical
maximum. By imposing the thermodynamical consistency between the pre- and
post-merger states through $\mathcal{I}_\mathrm{BBH}$, we showcase BRAHMA -- a
novel framework to infer the properties and astrophysical implications of
gravitational-wave detections. For GW190521 -- the heaviest confirmed binary
black hole merger observed so far -- our framework rules out high mass-ratio,
negative effective inspiral spin, and electromagnetic counterpart claims.
Furthermore, our analysis provides an independent confirmation that GW190521
belongs to a separate population.
