Phase diagrams for the Fiducial model at redshift z = 0, 1, 2, 5 for the original and smoothed fields (contours and density plot, respectively; lines and colours account for log-counts over five decades between 10 and 10⁵). As a reference, some polytropic laws are superposed to the P − n diagram at z = 0 (see legend) and the linear bias relation is indicated on the n_HI − n bivariate distribution at z = 0. Opaque symbols pinpoint threshold values where v₀ = 0.5 (star), v₂ = 0 (diamond), v₃ is maximum (triangle), minimum (dot circle), and has a second global maximum (filled circle), illustrating the relationship between thermodynamical and chemical states, possibly described by some scaling relation, and the average morphology of the corresponding excursion-set.