Strong Conformity and Assembly Bias: Towards a Physical Understanding of
  the Galaxy-Halo Connection in SDSS Clusters
  Understanding the physical connection between cluster galaxies and massive
haloes is key to mitigating systematic uncertainties in next-generation cluster
cosmology. We develop a novel method to infer the level of conformity between
the stellar mass of the brightest central galaxies~(BCGs) $M_*^{BCG}$ and the
satellite richness $\lambda$, defined as their correlation coefficient
$\rho_{cc}$ at fixed halo mass, using the abundance and weak lensing of SDSS
clusters as functions of $M_*^{BCG}$ and $\lambda$. We detect a halo
mass-dependent conformity as
$\rho_{cc}{=}0.60{+}0.08\ln(M_h/3{\times}10^{14}M_{\odot}/h)$. The strong
conformity successfully resolves the "halo mass equality" conundrum discovered
in Zu et al. 2021 -- when split by $M_*^{BCG}$ at fixed $\lambda$, the low and
high-$M_*^{BCG}$ clusters have the same average halo mass despite having a
$0.34$ dex discrepancy in average $M_*^{BCG}$. On top of the best-fitting
conformity model, we develop a cluster assembly bias~(AB) prescription
calibrated against the CosmicGrowth simulation, and build a conformity+AB model
for the cluster weak lensing measurements. Our model predicts that with a
${\sim}20\%$ lower halo concentration $c$, the low-$M_*^{BCG}$ clusters are
${\sim}10\%$ more biased than the high-$M_*^{BCG}$ systems, in excellent
agreement with the observations. We also show that the observed conformity and
assembly bias are unlikely due to projection effects. Finally, we build a toy
model to argue that while the early-time BCG-halo co-evolution drives the
$M_*^{BCG}$-$c$ correlation, the late-time dry merger-induced BCG growth
naturally produces the $M_*^{BCG}$-$\lambda$ conformity despite the well-known
anti-correlation between $\lambda$ and $c$. Our method paves the path towards
simultaneously constraining cosmology and cluster formation with future cluster
surveys.
