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.

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.