Stochasticity values for baseline HODs derived in alternate cosmologies (black points) and for different cylinder radii (magenta circles) and cylinder depths (blue squares). The tests for alternate cylinder sizes use the baseline HOD in the baseline Planck 2018 cosmology, and radii and depths in the legend are given in h⁻¹ Mpc. The 52 non-Planck cosmologies belong to the ABACUSSUMMIT emulator grid. All error bars are estimated via spatial jackknife resampling, and errors for radius 5 and depth 250 are of similar scale to the points as plotted. Dashed gray curves correspond to the results from Fig. 4 for reference, and the yellow star at the intersection of the curves indicates the stochasticity of the baseline HOD in the baseline cosmology. Dashed lines at α₀ = 1 and α₁ = 0 indicate the values corresponding to Poisson shot noise.