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A Capacity Criterion for Four-State Potts Reconstruction on Trees
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Abstract
For the ferromagnetic four-state broadcast model with $0\lt \lambda\lt 1$, we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances $\lambda^{-2|e|}$ is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Christopher D. Long, A Sharp Capacity Criterion for Heterogeneous Four-State Potts Broadcasting on Trees (GitHub preprint, 2026-10-11). Extends the two four-state papers to locally finite deterministic trees with edge-dependent eigenvalues satisfying $0<\lambda_e\le q<1$, claiming an exact $L^3$ capacity criterion whose necessity direction uses the threshold paper’s posterior moment-saving inequality. [family-level]
Elsewhere
- Emergent Mind: family 229
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026,
author = {{OpenAI}},
title = {{A Capacity Criterion for Four-State Potts Reconstruction on Trees}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf}{OAI:A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/229.md. The Lean statements have not been checked by people against the paper's claims.
Family 229: Exact three- and four-state reconstruction thresholds and four-state tree capacity
Proves the exact reconstruction threshold $d\lambda^2\gt 1$, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 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.
Other papers in this family:
- The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model · October 5, 2026
- The exact reconstruction threshold for the three-state symmetric channel · September 25, 2026