Pattern speed (top) and growth rate (bottom) of the dominant mode in Zang ν​ = ​4 disc (Zang 1976), as a function of the rescaled softening length, ε_F (Equation G.5), for the Plummer softening kernel (blue – Equation G.2) and the Kuzmin kernel (red – Equation G.3). The theoretical predictions (plain lines) are reproduced from DR+19b (Figure 3 therein), with the dashed line corresponding to the Newtonian (non-softened) prediction from Evans & Read (1998) (table D1b therein). The N–body measurements are from Sellwood & Evans (2001) (faint dots SE01 – Plummer softening only – Figure 1 panel b therein), and our own measurements (plain dots – both kernels). Here, we recover that the Plummer kernel has a strong gravity bias, i.e. a predicted change in the mode’s property $\mathrm{\Delta }{{\omega }_{\mathrm{M}}}^{\mathrm{Plummer}}\propto \epsilon$ as one varies the considered softening length, while the Kuzmin kernel satisfies $\mathrm{\Delta }{{\omega }_{\mathrm{M}}}^{\mathrm{Kuzmin}}\propto {\epsilon }^2$.