Beam-averaged line ratios between HCN(J = 1 → 0) (left) and N₂H⁺( J =1 → 0) (right) over ¹²CO( J =1 → 0) as a function of f_lense, for a sample of 100 000 N-PDFs with parameters uniformly sampled over N₀ ∈ [5 × 10²⁰ : 10²²], α ∈ [2.5 : 5], σ ∈ [0.2 : 1.3], and r_thres ∈ [2 : 100]. The dashed red line shows a linear relation, while the solid red line shows a linear fit to the data. This parameter space is more limited than the validity intervals of Table E.2, but more representative of N-PDFs observed in local clouds (Schneider et al. 2022).