# Root-free randomized graph construction research

Family 231: The free uniform spanning forest is a factor of IID. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Distributed-algorithm researchers can examine equivariant factor-of-IID representations for spanning forests and strongly Rayleigh processes.

## Applicability and commercial boundary

A measurable factor rule on an infinite graph need not have bounded local radius, practical termination or a finite distributed implementation.

## Initial business decision

Conditional research. Buyer budget, commercial novelty and profitability are unvalidated.

## Next verification action

Inspect coding radius, random-bit use and effective construction before proposing a distributed sampler.

## Evidence scope

The catalog statement was individually reviewed. This record does not imply a manuscript proof review or formal-scope comparison. No independent Lean check was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Main source

[The free uniform spanning forest is a factor of IID](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf).