{"id":"openai-math/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel","pdf":"https://github.com/openai/math/blob/main/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026","tags":["probability","reconstruction-problem","broadcast-on-trees","galton-watson-trees"],"score":0,"title":"The exact reconstruction threshold for the three-state symmetric channel","domain":"Probability and statistical mechanics","family":"229","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Exact three- and four-state reconstruction thresholds and four-state tree capacity"}