Result 236, Probability and statistical mechanics

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

Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when tanh⁡β≤(d−1)−1/2\tanh\beta\le(d-1)^{-1/2}, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.

Lean formalization Proof

The bigger picture

Why it matters

The Ising model describes random up-or-down spins that favor alignment with their neighbors. This manuscript identifies exactly when its free state on a regular infinite tree can be generated from independent randomness without choosing a root.

What changes?

For every integer d at least three, the tree has d neighbors per vertex. The manuscript reports that the free, zero-field state is a factor of IID exactly when the hyperbolic tangent of inverse temperature beta is at most one divided by the square root of (d minus one), for nonnegative beta, including equality. This means generating spins from independent, identically distributed vertex labels. 'Free, zero-field' means no imposed boundary spins or external field.

What does that help mathematicians do?

The positive direction, including the boundary case, is the new claim; impossibility above the threshold was already known. The proposed rule chooses no root and commutes with every tree symmetry on every label input, so it does not favor particular vertices or directions. Together, these statements pinpoint which states in this family can arise from independent randomness while preserving all tree symmetries.

Are there practical applications?

Its immediate value is foundational for probability and statistical mechanics: it separates spin states that admit a symmetry-preserving representation using independent randomness from those that do not. This is an exact representation result for an infinite system, not by itself a finite-time simulation algorithm or a guarantee that only nearby labels are needed.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The sharp factor-of-IID threshold for the free Ising model on regular trees

September 26, 2026 50 pages Main result formalized in Lean

For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when tanh⁡β≤(d−1)−1/2\tanh\beta\le(d-1)^{-1/2}. We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026,
  author = {{OpenAI}},
  title = {{The sharp factor-of-IID threshold for the free Ising model on regular trees}},
  howpublished = {OpenAI Math Release preprint
                  \href{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}{OAI:The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/236.md.

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

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result determines the factor-of-IID threshold for the free zero-field Ising law on the infinite dd-regular tree. For every d≥3d\ge3 and β≥0\beta\ge0, the law is a factor of IID exactly when tanh⁡β≤1/d−1\tanh\beta\le1/\sqrt{d-1}, including equality. The factor uses independent uniform vertex labels and is almost surely equivariant for each fixed tree automorphism. Finitary coding and coding-radius bounds are not included.

Comparator links

Result Comparator statement
Factor-of-IID threshold for the free Ising model FreeIsing.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.