  Gravitational wave detections are starting to allow us to probe the physical
processes in the evolution of very massive stars through the imprints they
leave on their final remnants. Stellar evolution theory predicts the existence
of a gap in the black hole mass distribution at high mass due to the effects of
pair-instability. Previously, we showed that the location of the gap is robust
against model uncertainties, but it does depend sensitively on the uncertain
$^{12}\rm{C}\left(\alpha,\gamma\right)^{16}\!\rm{O}$ rate. This rate is of
great astrophysical significance and governs the production of oxygen at the
expense of carbon. We use the open source MESA stellar evolution code to evolve
massive helium stars to probe the location of the mass gap. We find that the
maximum black hole mass below the gap varies between $40\rm{M}_\odot$ to
$90\rm{M}_\odot$, depending on the strength of the uncertain
$^{12}\rm{C}\left(\alpha,\gamma\right)^{16}\!\rm{O}$ reaction rate. With the
first ten gravitational-wave detections of black holes, we constrain the
astrophysical S-factor for
$^{12}\rm{C}\left(\alpha,\gamma\right)^{16}\!\rm{O}$, at $300\rm{keV}$, to
$S_{300}>175\rm{\,keV\, barns}$ at 68% confidence. With $\mathcal{O}(50)$
detected binary black hole mergers, we expect to constrain the S-factor to
within $\pm10$-$30\rm{\,keV\, barns}$. We also highlight a role for independent
constraints from electromagnetic transient surveys. The unambiguous detection
of pulsational pair instability supernovae would imply that
$S_{300}>79\rm{\,keV\, barns}$. Degeneracies with other model uncertainties
need to be investigated further, but probing nuclear stellar astrophysics poses
a promising science case for the future gravitational wave detectors.
