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 σ₈ of linear matter fluctuations inferred from rest-frame simulated dark matter halos in a comoving volume of 8\,(h⁻¹Gpc)³. Our constraint on σ₈ 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 kₘₐₓ= 0.10\,h\,Mpc⁻¹ (0.12\,h\,Mpc⁻¹), the field-level approach yields a factor of 1.7 (2.6) improvement on the σ₈ constraint, from 9.9% to 5.8% (9.4% to 3.6%). This study provides the first direct insights into cosmological information encoded in galaxy clustering beyond low-order n-point functions.