Trace coherence lemma for minimal walks (${\rho }_{\beta }\upharpoonright \alpha ={\rho }_{\gamma }\upharpoonright \alpha$)

If ${\rho }_{\beta }(\alpha )={\rho }_{\gamma }(\alpha )$, then the trace functions agree below $\alpha$. Each shorter trace is reconstructed by locating the first recorded club initial segment that meets $[\xi ,\alpha )$ and continuing the walk from its least such point.