Climbing the $N$-point Ladder Part I: Information in the Higher-Order Configuration-Space Clustering of Dark Matter Halos
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 $\{\Omega_m, \Omega_b, h, n_s, \sigma_8, M_\nu\}$ in real and redshift space from ${\sim}38{,}000$ GPU-accelerated $N$-point measurements. Treating the statistics as a ladder, $\mathrm{2PCF} \rightarrow +\mathrm{3PCF} \rightarrow +\zeta^{(4)}_{\mathrm{conn}}$, 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, by factors of $1.6$--$6.3$ over the redshift-space $\{\xi_0,\xi_2\}$ baseline with per-parameter gains consistent with those of the Fourier-space halo bispectrum on the same simulations. The 3PCF gains persist under conservative, modelling-motivated scale cuts: restricted to the tree-level-validated regime with minimum triangle side $\ge40\,h^{-1}{\rm Mpc}$, it still improves $M_\nu$ by $3.9\times$. The connected 4PCF adds a further $\sim1.2$--$1.5\times$, an increment insensitive to whether the quadrupole is included in the baseline but traceable to small-scale configurations with sides $\lesssim30\,h^{-1}{\rm Mpc}$. 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. The configuration-space ladder thus offers an independent and complementary route to the higher-order information probed by the Fourier-space poly-spectra.