OpenAI · 229-03 · 2026-09-25 · Probability and statistical mechanics
The exact reconstruction threshold for the three-state symmetric channel
preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026
Family 229. Sharp three- and four-state reconstruction thresholds.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
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Proves the exact reconstruction threshold , with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees () and observed Poisson trees (mean and , respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.
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 , 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 , and mean degree : recovery is possible exactly when . Above this threshold it is achievable in time; at or below it, weak recovery is information-theoretically impossible.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 229: Exact three- and four-state reconstruction thresholds and four-state tree capacity. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.