Correlation among halo mass M_500c at z = 0, formation redshift, and cool-coreness in TNG-Cluster, across cosmic epochs. The formation redshift throughout this paper is defined as the redshift at which a cluster reaches a mass of 10¹⁴ M_⊙. The upper panels show the correlation of formation redshift and the halo mass at z = 0. Symbols are color-coded by the central entropy of each halo at z = 0 (left plot) and at the formation redshift z_form (right panel), with the color bar indicating the classification into NCC, WCC, and SCC clusters. Histograms of the distribution of formation redshifts and cluster masses are shown on the top and right panels, respectively. The lower plots present entropy profiles at z = 0 (left) and at z_form (right) colored by core status. More massive halos form, i.e., reach the status of cluster, on average earlier than less massive halos. For a given halo mass, later forming halos are biased toward NCC states at z = 0. In addition, clusters form with low central entropy, with SCCs and WCCs having similarly shaped profiles exhibiting a prominent decline toward the core. According to TNG-Cluster, clusters are predominately born as CCs.