Is the free uniform spanning forest a factor of IID?

Theorem 1.1: there is one Borel, root-independent, equivariant rule $\Phi(G,U,e)$ such that for every infinite connected locally finite simple unweighted graph $G$ with IID uniform vertex labels $U$ , the edge set selected by $\Phi$ has the FUSF law. Further results: every translation-invariant strongly Rayleigh binary process on a countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, is a factor of IID, with no amenability needed. Not shown: finitary factors or weighted graphs beyond the strongly Rayleigh results.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Is the free uniform spanning forest a factor of IID? · FUSF as a factor of IID

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Is the free uniform spanning forest a factor of IID?

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No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Lyons's question as cited; the theorem gives one Borel, root-independent, isomorphism-equivariant rule for all infinite connected locally finite simple graphs, which covers Cayley graphs. Lean: the challenge FreeUniformSpanningForest (theorem OAI.Problem336.fusf_is_factor_iid, solution module OAI.Probability.SpanningForest.Factor) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. Graphs are encoded on vertex set N (so infinite), the FUSF law is defined by limits of uniform spanning tree cylinder probabilities over connected exhaustions, and the rule is required to be measurable and relabelling-equivariant. That is the headline. Not rebuilt here.