How much information can be extracted from galaxy clustering at the field level?

We present optimal Bayesian field-level cosmological constraints from nonlinear tracers of the large-scale structure, specifically the amplitude $\sigma_8$ of linear matter fluctuations inferred from rest-frame simulated dark matter halos in a comoving volume of $8\,(h^{-1}\mathrm{Gpc})^3$. Our constraint on $\sigma_8$ is entirely due to nonlinear information, and obtained by explicitly sampling the initial conditions along with bias and noise parameters via a Lagrangian EFT-based forward model, LEFTfield. The comparison with a simulation-based inference analysis employing the power spectrum and bispectrum -- likewise using the LEFTfield forward model -- shows that, when including precisely the same modes of the same data up to $k_{\mathrm{max}}= 0.10\,h\,\mathrm{Mpc}^{-1}$ ($0.12\,h\,\mathrm{Mpc}^{-1}$), the field-level approach yields a factor of 3.5 (5.2) improvement on the $\sigma_8$ constraint, from 20.0% to 5.7% (17.0% to 3.3%). This study provides direct insights into cosmological information encoded in galaxy clustering beyond low-order $n$-point functions.