What can galaxy clustering really tell us about the galaxy-halo connections?
Subhalo abundance matching (SHAM) is a commonly used framework for modeling the galaxy-halo connection. Yet, its standard implementation has difficulty reproducing the observed galaxy clustering with high accuracy (e.g., $\chi^2/\mathrm{dof} \approx 1$). To overcome this issue, we propose a novel CS-SHAM framework, in which central and satellite galaxies are independently matched to main and satellite subhalos in simulations. Within this scheme, we introduce three free parameters to explicitly characterize the satellite fraction, $f_{\mathrm{sat}}$, as a function of stellar mass or absolute magnitude. To evaluate the performance of CS-SHAM, we apply it to two sets of mock galaxy catalogs built with the conventional SHAM method but using different subhalo mass proxies, $M_{\mathrm{peak}}$ and $V_{\mathrm{peak}}$, as well as two additional galaxy samples generated from a SAM and from TNG-300. We demonstrate that CS-SHAM reliably reproduces galaxy clustering whether $M_{\mathrm{peak}}$ or $V_{\mathrm{peak}}$ is used as the subhalo mass proxy. We also find that the models are unable to place robust constraints on $f_{\mathrm{sat}}$ if different mass proxies are employed. Indeed, within the CS-SHAM framework the halo occupation distribution (HOD) and conditional luminosity or stellar mass function (CLF/CSMF) are accurately recovered. Furthermore, we demonstrate for the first time that galaxy clustering constrains the HOD and CLF/CSMF primarily for relatively massive halos. Because the halo bias is nearly constant for low-mass halos, galaxy clustering is generally not very sensitive to the satellite population residing in these low-mass systems.