Surface density perturbation of $\Sigma /\overline{\Sigma }$ (panel a) and log(Σ) (panel b), and specific torque density as a function of orbital distance (panel c) in run-fid for a Neptune-mass planet at t = 3.5 × 10⁴ P₀. The location of the planet is marked by a gray circle. In panel b, data are averaged over 20 snapshots in 1 P₀, the purple crosses indicate the wiggle torque positions, and the width between the vertical dashed lines is Δr_with = 2 max(r_H, r_hs). For visibility, the x-axis in panel c uses a linear scale up to 12 au and a logarithmic scale beyond that.