OpenAI · 229-02 · 2026-10-05 · Probability and statistical mechanics

A Capacity Criterion for Four-State Potts Reconstruction on Trees

preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-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.

For the ferromagnetic four-state broadcast model with $0 < λ < 1$ , we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances $λ^{- 2∣ e ∣}$ is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.

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.