Illustration of the variations in the first-order VSF across halos with distinct attributes. Each panel represents a specific property: relaxedness (top left), formation redshift (top right), SMBH mass (bottom left), and SMBH accretion rate (bottom right). The color scale quantifies the difference in VSF between relaxed and unrelaxed halos (top left), early and late-forming halos (top right), halos with high and low SMBH mass (bottom left), and halos with high and low SMBH accretion rates (bottom right). The y-axis denotes the halo mass, while the x-axis indicates the halo radial zone within which the VSF is measured. For each radial zone, the VSF is averaged over all separation scales as presented in Fig. 8. We note that averaging over all separation scales is justified, as we found the variation in Δlog₁₀(S₁) across these scales to be insignificant. The figure emphasizes that the VSF is sensitive to halo properties.