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 m, b, h, n s, sigma 8, M nu in real and redshift space from about 38, 000 GPU-accelerated N-point measurements. Treating the statistics as a ladder, 2PCF to + 3PCF to + zeta(4) conn, 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 about1.4 -- 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.