The exact reconstruction threshold for the three-state symmetric channel

We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when $d{\lambda }^2>1$, with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates $a,b>0$, and mean degree $(a+2b)/3>1$: recovery is possible exactly when $(a-b{)}^2>3(a+2b)$. Above this threshold it is achievable in $O(n\log n)$ time; at or below it, weak recovery is information-theoretically impossible.