Lemma. Pairing preserves the declared marginal target [ftip-00GY]
✍️source

    Under a fixed nonadaptive slice and the iid coupling of Definition [ftip-00FE], the paired estimator remains unbiased for the two marginal scores in Theorem [ftip-00FF].

    Proof. 

      Condition on the fixed slice. The coupling has the stated marginals, so the proof of Theorem [ftip-00FF] applies without changing either expectation.