Influence of BP/X-shaped structures and dark matter halos on model results. The top panels show the marginalised chi-squares χ_norm² = (χ² − χ_min²)/Δχ_CL² versus model-recovered bar pattern speeds Ω_p, while the bottom panels display χ_norm² versus model-recovered bar azimuthal angles φ. Left panels correspond to the results of Au-23-85-50 and right panels are for Au-23-85-80. Each fold line represents a complete set of modelling runs. The black fold lines denote the default model results, which incorporate BP/X-shaped structures and allow dark matter halos to vary freely, as previously displayed in the middle panel of Fig. 7. The blue lines correspond to results with constrained dark matter halos, requiring the differences between model-recovered and true dark matter mass within 10 kpc to be less than 10% ($0.9M_{\mathrm{dm},10\mathrm{kpc}}^{\mathrm{true}}<M_{\mathrm{dm},10\mathrm{kpc}}^{\mathrm{model}}<1.1M_{\mathrm{dm},10\mathrm{kpc}}^{\mathrm{true}}$). The red lines represent the results ignoring BP/X-shaped structures. The horizontal dashed lines represent 1σ (68%, χ_norm² = 1) and 2σ (95%, χ_norm² = 2) confidence levels. The vertical dotted lines denote the true bar pattern speeds Ω_T or azimuthal angles φ_T, with their values shown by the green annotations.