The Nam-Sly-Zhang conjecture: the free Ising state on the d-regular tree is a factor of IID up to the reconstruction threshold

Theorem 1.1: for every $d\ge3$ and $\beta\ge0$, $\mu_{d,\beta}$ is a factor of IID if and only if $\tanh\beta\le(d-1)^{-1/2}$; the new part is the positive implication including equality, and the factor can be chosen to commute with every automorphism on every input. The converse is Sly's, as given by Lyons, with the Backhausz-Szegedy-Virag bound. The construction inverts the innovations of the Nam-Sly-Zhang stochastic-localization observation process. It does not treat antiferromagnetic or nonzero-field states, other Gibbs measures, or finitary factors.

Theorem 1.1: for every $d\ge3$ and $\beta\ge0$, $\mu_{d,\beta}$ is a factor of IID if and only if $\tanh\beta\le(d-1)^{-1/2}$; the new part is the positive implication including equality, and the factor can be chosen to commute with every automorphism on every input. The converse is Sly's, as given by Lyons, with the Backhausz-Szegedy-Virag bound. The construction inverts the innovations of the Nam-Sly-Zhang stochastic-localization observation process. It does not treat antiferromagnetic or nonzero-field states, other Gibbs measures, or finitary factors.