  We present the first step towards deriving cosmological constraints through
the abundances of galaxy clusters selected in a $510\,\mathrm{deg}^2$
weak-lensing aperture mass map, constructed with the Year-Three shear catalog
from the Hyper Suprime-Cam Subaru Strategic Program. We adopt a conservative
source galaxy selection to construct a sample of $129$ weak-lensing peaks with
a signal-to-noise ratio above $4.7$. We use semi-analytical injection
simulations to derive the selection function and the mass--observable relation
of our sample. These results take into account complicated uncertainties
associated with weak-lensing measurements, such as the non-uniform survey depth
and the complex survey geometry, projection effects from uncorrelated
large-scale structures, and the intrinsic alignment of source galaxies. We also
propose a novel modeling framework to make parts of the mass--observable
relation insensitive to assumed cosmological parameters. Such a framework not
only offers a great computational advantage to cosmological studies, but can
also benefit future astrophysical studies using shear-selected clusters. Our
results are an important step towards utilizing these cluster samples that are
constructed nearly independent of any baryonic assumptions in upcoming
deep-and-wide lensing surveys from the Vera Rubin Observatory, Euclid, and the
Nancy Grace Roman Space Telescope.
