= Alternating exterior-arm separation lemma
Four infinite exterior arms entering a square in alternating primal-open and dual-open order force the two primal arms to be disjoint outside the square. If they met, their finite connecting crosscut together with a boundary arc would enclose one dual arm, preventing its escape to infinity. Connecting the two primal arms through the square creates a proper doubly infinite <open path>, separating the two remaining dual infinite tails. <finite-energy property of Bernoulli percolation> can then turn this configuration into two infinite dual clusters, contradicting uniqueness. Care is needed to preserve the exterior dual tails even if the finite modification destroys their original boundary attachments.
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