Results of the HOD Monte Carlo sampling procedure described in Sect. 4.4. Separate searches were performed with assembly bias (blue points) and without (i.e., all assembly bias parameters set to zero, red points) in the baseline cosmology. The HODs shown achieve the target values of linear galaxy bias (1.5) and density (10⁻³ h³ Mpc⁻³) to within 2 percent and satisfy the additional constraint that the maximum of ⟨N_g⟩ across all halos falls in the range [25, 50], as specified in Eq. (14). The values of stochasticity (α₀, α₁) and quadratic bias (b₂) shown here are the mean of measurements on 20 mock galaxy catalogs generated by each HOD, as in Fig. 6. The lower-dimensional search without assembly bias was allowed to run until 500 such HODs were sampled; the search including assembly bias was run until 200 HODs were found. Parameter ranges for the sampling procedure are given in Table A.1.