Comparison between the transition parameter, α, and the critical value, α_crit(γ, R_e/R_b, m), that separates physically admissible and inadmissible core–Sérsic models. The horizontal axis shows the fitted approximation α_crit, fit(γ, R_e/R_b, m), while the vertical axis shows the actual transition parameter, α. The red line indicates the boundary α = α_crit: models below this line have monotonically decreasing intrinsic density profiles and are therefore physically admissible, whereas models above the line produce non-monotonic densities and are unphysical. The colour scale indicates the inner logarithmic slope, γ. Overplotted are core–Sérsic parameters from the observational studies of Richings et al. (2011, blue squares) and Dullo & Graham (2013, violet stars), illustrating that most observed systems lie within the physically allowed region, while a few fall in the inadmissible part of the parameter space.