Uniform even subgraphs and graphical representations of Ising as factors of i.i.d.

The candidate solution reproduces the paper’s construction: couple Loop O(1) to FK–Ising with p=2x/(1+x), ph=2y/(1+y); sample FK–Ising as a (graph) factor of i.i.d.; on each cluster build, as a factor, a one-ended spanning forest; extract a locally finite generating family of the relevant cycle space; assign i.i.d. fair bits and take the mod-2 sum to sample the uniform even subgraph; compose these factor maps. It also correctly treats the exceptional (x,y)=(1,0) two-ended case and the free-boundary cases (y>0, invariant amenable, geodesic-cycle condition, planar maps). These steps align with Theorems 1.1–1.2 and supporting lemmas/propositions in the paper, with no substantive gaps or contradictions.