Remark: 229-03. The exact reconstruction threshold for the three-state symmetric channel

Authors: OpenAI. Family 229.

Title: The exact reconstruction threshold for the three-state symmetric channel.

Abstract: 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 $d{}lambda{}2{}gt 1$, 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 $a,b{}gt 0$, and mean degree $(a+2b)/3{}gt 1$: recovery is possible exactly when $(a-b){}2{}gt 3(a+2b)$. Above this threshold it is achievable in $O(n{}log n)$ time; at or below it, weak recovery is information-theoretically impossible.

This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.