# The exact factor-of-IID threshold for free Ising spins on trees

> The free Ising state on a regular tree can be built from independent labels exactly up to tanh β = 1/√(d−1).

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-236/
Last updated: 2026-10-14

**The free Ising state on a regular tree can be built from independent labels exactly up to tanh β = 1/√(d−1).**

**Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 26, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 236

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Put plus or minus spins on an endless branching tree, with neighbors tending to agree. The question is whether a fixed rule can make these spins from independent random labels, with no starting vertex. The paper claims this works exactly up to a known cut-off, including the cut-off itself. Past the cut-off it was already known to fail.

## At a glance

- **First step:** 2022 (Danny Nam, Allan Sly and Lingfu Zhang, factor for large d and small θ)
- **Significance:** Solid. It settles the Nam–Sly–Zhang conjecture on the exact range, including the critical point.
- **Trust:** Formal proof listed
- **Field:** Probability and statistical mechanics (1 paper)

## The explanation

Take a tree where every point, called a vertex, has exactly three neighbors, and the branching never stops. Put a spin, +1 or −1, on every vertex. Start at one vertex with a fair coin. Each neighbor then copies its parent's spin with probability (1 + θ)/2, and flips it otherwise. Keep going outward along every branch. Here θ = tanh β, and β is the inverse temperature. A large β means spins agree more often.

Try θ = 0.7. A child agrees with its parent 85% of the time. Two spins n steps apart have correlation θ to the power n. So at distance 10 the correlation is 0.7^10, about 0.028.

This random picture is the free Ising state on the tree. Its rules mention a starting vertex, but the final law does not depend on which one. The question is whether you can build it with no starting vertex at all. You give every vertex its own independent random number, a label. Then one fixed rule turns all the labels into spins. The rule must treat every vertex alike: rotate or reflect the tree, and the output moves with it. Such a rule is called a factor of IID, short for independent, identically distributed labels.

The paper studies trees where every vertex has d neighbors, with d at least 3. It claims a factor of IID exists exactly when tanh β ≤ 1/√(d−1)[1]. For d = 3 that cut-off is θ = 1/√2, about 0.7071. So θ = 0.7 is just inside, and θ = 0.72 is just outside.

Some of this was known. Past the cut-off, no factor exists, by an argument of Sly recorded by Lyons[2]. Well below it, a simpler construction from finite clusters works[3]. In 2022, Nam, Sly and Zhang built a factor for large d and small θ. Their method used random equations driven by independent Brownian motions, a model of continuous random noise. They left the full range as a conjecture. The paper claims the positive direction, including the cut-off itself[4]. So it claims to settle their conjecture for every d ≥ 3[5].

The cut-off has a second meaning. It is the point where distant spins stop carrying information about one spin[6]. But that fact alone does not give a rule built from labels[7].

The paper follows the observation method of Nam, Sly and Zhang. Imagine watching every spin through noise that slowly fades. Remove the part of what you see that you could already predict. What remains is fresh noise, called innovations, independent at each vertex[8]. Labels can produce such noise directly. The hard step is to show that the innovations alone determine the spins.

The paper's test runs two copies of the observations with the same innovations. It shows the copies must agree. Below the cut-off, a size-weighted sum shrinks fast at each distance, and that gives agreement easily. At the cut-off it does not shrink at all[9]. For d = 3 there are 3 × 2^(n−1) vertices at distance n. Multiply by θ^(2n) with θ = 1/√2. You get 1.5 at every distance n. The paper replaces the lost decay with three new estimates[10]. They end in a growth inequality that forces the gap between the copies to be zero[11].

The result has limits. It does not say how many labels decide one spin[12]. It gives no bounds that hold across all d and β at once[13].

How sure is this? Lean, a computer proof checker, has a formal version of the main statement. It covers every d ≥ 3 and β ≥ 0, including equality at the cut-off. The scope note says finitary coding and coding-radius bounds are not included. No specialist review of the paper has been published.

**What it doesn't show.** This does not say how many labels are needed to decide one spin.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Solid if the proof holds. Lean has a formal version of the threshold statement; no specialist review is published, and no coding-radius bound is given.

## Words used

- **Factor of IID:** A fixed rule that turns independent random labels at every vertex into the desired random pattern, treating all vertices alike.
- **d-regular tree:** An endless tree in which every vertex has exactly d neighbors.
- **Free Ising state:** Random ±1 spins made by copying from a vertex outward, each step agreeing with probability (1 + θ)/2.
- **Inverse temperature β:** A strength setting: larger β makes neighboring spins agree more often.
- **Innovations:** The part of a noisy observation that could not be predicted from what was seen before.
- **Reconstruction:** Guessing one spin better than chance from spins very far away.

## The original paper

- [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf) - September 26, 2026 · Formal proof listed

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/236.md)

## Quoted lines from the paper (13)

[1] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> For every integer \(d\ge3\) and inverse temperature \(\beta\ge0\), the free zero-field ferromagnetic Ising measure on the infinite \(d\)-regular tree is a factor of IID exactly when \(\tanh\beta\le(d-1)^{-1/2}\).

[2] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> The strict converse is known: the obstruction beyond the reconstruction threshold is due to Sly, as recorded and proved in Lyons.

[3] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> In the uniqueness range $\theta\le b^{-1}$, including equality, a finite-cluster construction gives the factor; see Lyons.

[4] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> The construction proves the positive implication, including equality.

[5] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> Theorem … proves their conjecture for every $d\ge3$, including equality and their pointwise convention for equivariance.

[6] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> For the free state on $T_d$, reconstruction from distant levels is possible exactly when $b\theta^2>1$; equivalently, the free Gibbs state is extremal exactly when $b\theta^2\le1$.

[7] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> Nonreconstruction supplies no such rule, and a weak limit of factor laws need not itself be a factor.

[8] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> These innovations are independent Brownian motions in the observation filtration.

[9] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> At $b\theta^2=1$ those two factors cancel exactly.

[10] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> Three estimates replace the lost geometric decay.

[11] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> The complete noise and drift calculation then gives an integrable Osgood-type inequality for $V$, forcing $V=0$.

[12] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> Measurability does not require a spin to be determined by finitely many labels; no finitary coding or coding-radius estimate is asserted.

[13] From [The sharp factor-of-IID threshold for the free Ising model on regular trees](https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf):

> The map is obtained for each fixed $d$ and $\beta$ in the permitted range; no parameter-uniform or computational bound is asserted.

## Other sources

- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”
- [D. Nam, A. Sly and L. Zhang, “Ising Model on Trees and Factors of IID”, Communications in Mathematical Physics 389(2) (2022), 1009–1046](https://doi.org/10.1007/s00220-021-04260-2)
- [R. Lyons, “Factors of IID on Trees”, Combinatorics, Probability and Computing 26(2) (2017), 285–300](https://doi.org/10.1017/S096354831600033X)
- [Á. Backhausz, B. Szegedy and B. Virág, “Ramanujan graphings and correlation decay in local algorithms”, Random Structures & Algorithms 47(3) (2015), 424–435](https://doi.org/10.1002/rsa.20562)

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