VibeMathedMath problems solved with AI
Sign in

The Nam-Sly-Zhang conjecture: the free Ising state on the d-regular tree is a factor of IID up to the reconstruction threshold

Let TdT_d be the infinite dd-regular tree, d≥3d\ge3, and μd,β\mu_{d,\beta} the free zero-field ferromagnetic Ising measure with θ=tanh⁡β\theta=\tanh\beta. A spin law is a factor of IID if it is the law of Φ(U)\Phi(U) for IID uniform vertex labels UU and a measurable Φ\Phi commuting with tree automorphisms. Lyons (2017) studied which such Gibbs measures are factors of IID: the free state is one when θ≤1/(d−1)\theta\le1/(d-1), and Sly's argument shows it is not when θ>(d−1)−1/2\theta>(d-1)^{-1/2}, beyond the reconstruction threshold. Nam, Sly and Zhang (arXiv 2020, CMP 2022) proved it is a factor for θ≤c/d−1\theta\le c/\sqrt{d-1} and d≥d0d\ge d_0 and conjectured the sharp threshold. Is μd,β\mu_{d,\beta} a factor of IID for every d≥3d\ge3 whenever tanh⁡β≤(d−1)−1/2\tanh\beta\le(d-1)^{-1/2}, including equality?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Ergodic theory on trees; Ising model, factors of IID
Posed by
D. Nam, A. Sly and L. Zhang, Ising model on trees and factors of IID (arXiv 2012.09484, 2020; Comm. Math. Phys. 2022), following R. Lyons, Factors of IID on trees (2017)
Year posed
2020
Years open
6y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
16 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every d≥3d\ge3 and β≥0\beta\ge0, μd,β\mu_{d,\beta} is a factor of IID if and only if tanh⁡β≤(d−1)−1/2\tanh\beta\le(d-1)^{-1/2}; the new part is the positive implication including equality, and the factor can be chosen to commute with every automorphism on every input. The converse is Sly's, as given by Lyons, with the Backhausz-Szegedy-Virag bound. The construction inverts the innovations of the Nam-Sly-Zhang stochastic-localization observation process. It does not treat antiferromagnetic or nonzero-field states, other Gibbs measures, or finitary factors.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the Nam-Sly-Zhang conjecture. The family is in lean/formalization.yaml (ComparatorChallenges/FreeIsing.json, declaration OAI.Problem367.free_ising_factor_iff_threshold, file OAI/Probability/FreeIsing/Main.lean). The comparator statement was read here: for every d≥3d\ge3 and β≥0\beta\ge0, there is a measurable map of IID labels, almost surely equivariant for each fixed automorphism, whose image law matches the free Ising cylinder weights on connected finite sets, if and only if tanh⁡β≤1/d−1\tanh\beta\le1/\sqrt{d-1}. This is the headline, including the known converse. Not rebuilt here. Finitary coding and coding-radius bounds are not claimed, and only the ferromagnetic free state is treated.

Sources

Changelog1 change

Discussion