This work presents the fourth paper of the CoreLumen Gravity series. Starting from the modified gravitational response functions derivedin CoreLumen Gravity II, we extend the analysis of condensate gravityinto the nonlinear regime of cosmic structure formation. The modified Poisson equation k²Φ = −4πGa² µ(k,z) ρ_m δ alters the gravitational force responsible for matter clustering.We investigate the resulting impact on nonlinear growth,spherical collapse, halo formation, and the evolution of thematter power spectrum. The modified response propagates into the nonlinear sector,changing the collapse threshold, the halo mass function,and the statistical distribution of matter fluctuations. Within this framework the abundance of halos can be describedusing a Press–Schechter–type approach, where the halo massfunction depends on the modified variance of the density fieldand the condensate-modified gravitational response. The resulting structure-formation framework provides thenonlinear cosmological counterpart to the perturbativeresponse theory developed in Gravity II and establishesthe theoretical basis for connecting condensate gravityto observational large-scale structure statistics. This paper therefore forms the nonlinear bridge betweenthe covariant condensate gravity framework and theobservational analyses developed in the CoreLumen COS program.