OpenAI · 229-01 · 2026-10-05 · Probability and statistical mechanics

The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model

preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026

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

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the exact reconstruction threshold $d λ^{2} > 1$ , with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees ( $d \geq 2$ ) and observed Poisson trees (mean $d > 1$ and $d > 0$ , respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of $λ$ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.

We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when $d λ^{2} > 1$ ; we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival. The proof uses reproducible exact-arithmetic verification of polynomial inequalities.

Preprint source and Lean scope

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A Lean scope note is stored for family 229, but it does not name this preprint. It names 229-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.