Number of core transformations across the history of each TNG-Cluster halo, as a function of M_500c at z = 0. In the main panel, black dots represent individual halos, while the equal-width violins depict the corresponding mass distributions. The black crosses show the median halo mass for each number of transformations. The subpanels show the distributions of number of transformations (top) and halo mass (right). In the top, we contrast our fiducial analysis choice (dark blue) with the impact of varying the threshold applied to the K₀ time derivative (dark green and light green histograms). The histogram in the right panel shows the mass distribution for clusters that have only one (equal or more than three) transformation(s) in darker (lighter) green. In general, the majority of clusters undergo between one and three transformations over their lifetime. More massive clusters undergo more transformations, on average.