The free uniform spanning forest as a factor of IID
From OpenAI Aftermath
The free uniform spanning forest as a factor of IID is the question whether the free uniform spanning forest of an infinite graph can be generated from independent random labels on its vertices by one measurable rule that treats all vertices alike. A manuscript dated 25 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a single such rule for every infinite connected locally finite simple graph, and a factor-of-IID representation of every translation-invariant strongly Rayleigh process on a countable group. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Pemantle constructed infinite-volume limits of uniform spanning trees on the integer lattice, and Benjamini, Lyons, Peres and Schramm developed the general theory. The free uniform spanning forest is the weak limit of uniform spanning trees on finite connected exhaustions, without identifying boundary vertices.
- On transient graphs the wired forest has an IID construction through Wilson's algorithm rooted at infinity, and Angel, Ray and Spinka gave an explicit graph-factor formulation. Lyons and Thom proved Bernoulli isomorphism for equivariant determinantal measures on amenable Cayley graphs. Lyons discussed the factor question for the free forest on Cayley graphs in his report for a 2013 Oberwolfach workshop.
- Timár proved that the free forest is a factor of IID on recurrent and, more generally, invariantly amenable unimodular random graphs, and his December 2025 version states the unrestricted question as open.
The original result
- Claim. Theorem 1.1 of The free uniform spanning forest is a factor of IID states that there is one Borel, root-independent rule, equivariant under every isomorphism of labeled graphs, which turns independent uniform vertex labels on any infinite connected locally finite simple unweighted graph into an edge set with exactly the law of the free uniform spanning forest. A corollary gives a graph factor of IID under every unimodular law on these graphs. No amenability, transience, degree bound or degree moment is assumed, and the factor is not claimed to be finitary. A second theorem states that every strongly Rayleigh law on the subsets of a countable group that is invariant under translations is a factor of IID for the group action, including invariant determinantal processes with Hermitian positive-contraction kernels; no amenability is needed.
- Method, as described: each edge indicator is observed through its own Brownian noise. A response identity for finite weighted spanning trees, in which the total sensitivity of an edge probability to all edge weights equals 2p(1 − p) ≤ 1/2, passes to infinite volume and makes the posterior drift uniformly Lipschitz. The innovations are independent Brownian motions, a Picard iteration inverts the observation equation for every continuous input, and independent vertex labels supply the edge noises by a local allocation without a root. The manuscript names stochastic localization, as developed by Eldan and by El Alaoui and Montanari, and the factor codings of Nam, Sly and Zhang for free Ising measures on regular trees as antecedents.
- Source (openai/math repository, commit
7ef5c0c): The free uniform spanning forest is a factor of IID, dated 25 September 2026. - Formalization, as described by the repository. The scope note lean/docs/231.md states that the formalization proves the universal root-independent factor theorem for the free forest, and that linked supplements give equivariant IID sampling for invariant strongly Rayleigh laws on countable groups and existence, uniqueness and factor results for determinantal laws with Hermitian positive-contraction kernels on countable index sets. It lists the Comparator statements
FreeUniformSpanningForest.leanandStronglyRayleighDPP.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 29 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | Sampling the free uniform spanning forest from IID labels, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the claimed graphwise sampling theorem through a finite response identity, Gaussian observations with independent innovations, the construction of an everywhere-defined drift, a decoder that is solved for every input, and independent edge noise allocated without a root. It notes that the rule is not asserted to inspect only finitely many labels. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit 94c6630, 28 October 2026) |
Review (fidelity of the formal statements) | Rates this family "full": the two Lean statements cover both claims of the summary, the single equivariant rule for every graph and the strongly Rayleigh and determinantal factors. It records sandboxed Comparator passes for both challenges on 8 October 2026. | MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K, run 38051470892: FreeUniformSpanningForest and StronglyRayleighDPP | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept both solutions, in 572 s and 341 s. The project reports two-kernel certification of 398 of 416 challenges of the collection. | Repository. |
| 2026-10-10 to 2026-10-14 | con-leche development journal (task #348) | Verification (second kernel) | In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions lists neither challenge of this family, so both were accepted by con-leche and by Lean's official kernel. |
Development journal. |
| 2026-10-12 | The free uniform spanning forest, a course on Open Mathematics Courses (kokunoyumeto/open-math-courses-public) | Reading (exposition) | Four lessons: weighted spanning trees, the forest as a limit of trees, tilted laws and the drift, and the integral equation of the drift. The course describes itself as the start of an exposition of the theorem and states that the factor-of-IID theorem itself is not yet included. | The course states that it was written and self-checked by Claude Opus 5.5 (Anthropic) and that independent review is not complete. |
| 2026-10-17 | AI Proved What?, episode 231 Can coin flips grow a forest?, remade version of 22 October 2026 | Reading (exposition) | Video explainer of the claim of one rule that turns independent vertex labels into the free uniform spanning forest on every infinite connected locally finite simple graph, and of the strongly Rayleigh result on countable groups. The episode states that a finitary rule, an isomorphism with a Bernoulli shift, weighted graphs, and the fixed-price problem in group theory are not claimed. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 19 pages, dated September 25, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: yes. The page also states that the site scanned about 330 Lean files without building them and found no unproved steps, that OSVerify has not reviewed the result, and that it found no public reaction. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
