Nonuniqueness of site percolation on nonamenable quasi-transitive graphs

Benjamini and Schramm conjectured in 1996 that site percolation on every nonamenable quasi-transitive graph has p_c < p_u. We prove this in Hutchcroft's operator form: for Bernoulli site percolation on every connected, locally finite, nonamenable quasi-transitive graph, the critical two-point operator is bounded on ℓ², p_c < p_{2→2} ≤ p_u, and in the standard monotone coupling infinitely many infinite clusters coexist at every parameter of a nondegenerate interval. Since bond percolation is site percolation on the line graph, this contains OpenAI's recent bond theorem. For unimodular automorphism groups we adapt OpenAI's proof: cut vertices replace bridges, a pruning lemma handles retained contact vertices, a rejection step controls the junctions created by opening a vertex, and pivotal corridors become induced paths. For nonunimodular groups we extend Tran and Nguyen's site argument from vertex-transitive to quasi-transitive graphs.