Constituent-Tagged Transfer Linking Cluster Dynamics to Static Scaling

Clusters that merge and split have no unique genealogy. Tracking the host size of a conserved constituent defines $q_{\ast }$ by $⟨(S_{t+\tau }\mathrm{/}{S}_t{)}^{q_{\ast }}⟩=1$, with $q_{\ast }\mathrm{\ne }0$. Across coupled map networks, coagulation-fragmentation dynamics, and Vicsek systems, $q_{\ast }\simeq \alpha -2$ without fitting, where $c(s)\propto s^{-\alpha }$ on the same size window. Radius scans locate the known Vicsek interaction radius and recover an independently specified clustering scale in experimental active rods. Local scale covariance yields the relation, while a tent map counterexample shows when reducing the stationary balance at each parent size to one moment fails.