  Empirical studies of the relationship between baryonic matter in galaxies and
the gravitational potential of their host halos are important to constrain our
theoretical framework for galaxy formation and evolution. One such relation,
between the atomic hydrogen (HI) mass of central galaxies ($M_{\rm{HI,c}}$) and
the total mass of their host halos ($M_{\rm{halo}}$), has attracted significant
interest in the last few years. In this work, we use the extended GALEX Arecibo
SDSS Survey to examine the scatter of the HI-halo mass relation for a
representative sample of central galaxies. Our findings reveal a flat median
relation at $\rm{log}_{10}$$(M_{\rm{HI,c}}/\rm{M}_{\odot}) \approx 9.40$,
across $11.1 < \rm{log}_{10}$$(M_{\rm{halo}}/\rm{M_{\odot}}) < 14.1$. This flat
relation stems from the statistical dominance of star-forming, disc galaxies at
low $M_{\rm{halo}}$ in combination with the increasing prevalence of passive,
high stellar-concentration systems at higher $M_{\rm{halo}}$. The scatter of
this relation and the stellar specific angular momentum of centrals have a
strong link (Spearman's rank correlation coefficient $\geq 0.5$). Comparisons
with simulations suggest that the kinematic state of host halos may be
primarily driving this scatter. Our findings highlight that the HI-halo mass
parameter space is too complex to be completely represented by simple median or
average relations and we show that tensions with previous works are most likely
due to selection biases. We recommend that future observational studies, and
their comparisons with theoretical models, bin central galaxies also by their
secondary properties to enable a statistically robust understanding of the
processes regulating the cold gas content within central galaxies of
dark-matter halos.
