Climbing the $N$-point Ladder Part I: Information in the Higher-Order Configuration-Space Clustering of Dark Matter Halos

The two-point correlation function completely describes a Gaussian random field, but nonlinear gravitational growth, halo bias, and redshift-space distortions drive the late-time halo field strongly non-Gaussian, moving a substantial part of the cosmological information into higher-order correlations. We quantify the information content of the configuration-space two-, three-, and connected four-point correlation functions of Quijote dark-matter haloes at $z=0$ and fixed number density. We build Fisher forecasts for $\{{\mathrm{\Omega }}_m,{\mathrm{\Omega }}_b,h,n_s,{\sigma }_8,M_{\nu }\}$ in real and redshift space from $\sim \mathrm{38,000}$ GPU-accelerated $N$-point measurements. Treating the statistics as a ladder, $2\mathrm{P}\mathrm{C}\mathrm{F}\to +3\mathrm{P}\mathrm{C}\mathrm{F}\to +{\zeta }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}^{(4)}$, we report the information gained at each rung. The 3PCF supplies most of the accessible higher-order information: it tightens every parameter, most strongly ${\sigma }_8$ and $M_{\nu }$, whose degeneracy it partially breaks, with per-parameter gains consistent with those of the Fourier-space halo bispectrum on the same simulations. The connected 4PCF adds a further $\sim 1.4$–$1.5\times$. This rung-to-rung increment is stable against derivative-sample noise and compression regularization, whereas the absolute constraints remain limited by the finite simulation ensembles and are reported as preliminary. We validate the measured 3PCF against a tree-level perturbation-theory model, recovering a linear bias consistent with the 2PCF. The configuration-space ladder thus offers an independent and complementary route to the higher-order information probed by the Fourier-space poly-spectra.