The Kesten–Stigum bound is tight for the four-state Potts model on the binary tree

Consider the broadcast of a uniformly random colour from the root of the infinite binary tree, where each edge copies its parent's colour with probability $\lambda+(1-\lambda)/4$ and otherwise chooses one of the other three colours uniformly. We prove that for $0<\lambda\le 1/\sqrt2$ the colours at depth $n$ carry asymptotically no information about the root colour. Since reconstruction is possible for $2\lambda^2>1$ by the theorem of Kesten and Stigum, the Kesten–Stigum bound is the reconstruction threshold for the ferromagnetic four-state Potts channel on the binary tree. The proof is computer assisted. A certified upper bound on the second moment of the root posterior, obtained from an exact finite-state upgrade of the belief-propagation recursion, places the limiting posterior law in a small neighbourhood of the uniform law; an exact polynomial certificate then shows that no nontrivial belief-propagation fixed point lies in that neighbourhood.