Comparison of the mode frequencies computed using the method $N_{\mathrm{BV}}^2=0$ with the frequencies of the mixed modes of lowest inertia in each Δν interval, without modifying the outputs of MESA. (a) Model frequencies of ℓ = 1 modes in units of the large frequency separation Δν for the first eight radial orders and for a 1 M_⊙ track during the He-core burning and the early He-shell burning phase (input physics are summarised in Table 1). The dipole modes computed with the method $N_{\mathrm{BV}}^2=0$ are shown in blue circles, while the mixed dipole modes of lowest inertia are indicated by blue crosses. The dotted grey line shows the end of the clump phase, defined as the point when the central helium mass fraction was below 0.01. (b) Histogram of the differences between the dipole mode frequencies ν_1,  p computed with the method $N_{\mathrm{BV}}^2=0$ (blue circles) and the frequencies ν_1,  g of the dipole modes of lowest inertia (blue crosses). This histogram is computed for models in the entire He-burning phase and all radial orders up to ν/Δν = 9. These differences are expressed as a percentile of Δν. The solid black line localises the median of the distribution, while the dotted lines show the 16^th and 84^th percentiles of the distribution. (c) and (d) Same label as the upper panels, but for the ℓ = 2 modes. Some modes could not be computed because they were missing, or because they were inconsistent. This explains why some symbols are missing, especially the mixed non-radial modes of lowest inertia at ν/Δν ≃ 1.