COSMOS-Web: stellar mass assembly in relation to dark matter halos across $0.2<z<12$ of cosmic history

We study the stellar mass function (SMF) and the co-evolution with dark matter halos via abundance matching in the largest redshift range to date $0.210.5$) implying integrated star formation efficiencies (SFE) $\epsilon_{\star}\equiv M_{\star}\, f_{\rm b}^{-1} M_{\rm halo}^{-1} \gtrsim 0.5$. We find a flattening of the SMF at the high-mass end that is better described by a double power law at $z>5.5$. At $z \lesssim 5.5$ it transitions to a Schechter law which coincides with the emergence of the first massive quiescent galaxies in the Universe. We trace the cosmic stellar mass density (SMD) and infer the star formation rate density (SFRD), which at $z>7.5$ agrees remarkably with recent \JWST{} UV luminosity function-derived estimates. However, at $z \lesssim 3.5$, we find significant tension ($\sim 0.3$ dex) with the cosmic star formation (SF) history from instantaneous SF measures, the causes of which remain poorly understood. We infer the stellar-to-halo mass relation (SHMR) and the SFE from abundance matching out to $z=12$, finding a non-monotonic evolution. The SFE has the characteristic strong dependence with mass in the range of $0.02 - 0.2$, and mildly decreases at the low mass end out to $z\sim3.5$. At $z\sim3.5$ the SFE increases sharply from $\sim 0.1$ to approach high SFE of $0.8-1$ by $z\sim 10$ for log$(M_{\rm h}/M_{\odot})\approx11.5$, albeit with large uncertainties. Finally, we use the SHMR to track the SFE and stellar mass growth throughout the halo history and find that they do not grow at the same rate -- from the earliest times up until $z\sim3.5$ the halo growth rate outpaces galaxy assembly, but at $z>3.5$ halo growth stagnates and accumulated gas reservoirs keep the SF going and galaxies outpace halos.