A Capacity Criterion for Four-State Potts Reconstruction on Trees

For the ferromagnetic four-state broadcast model with $0<\lambda <1$, we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L${}^3$ capacity with edge resistances ${\lambda }^{-2|e|}$ is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.