Stability of Change-of-Num\'eraire Reweighting: An Exact Wasserstein Boundary

Reweighting a probability law by a positive num\'eraire and pushing forward the payoff-to-num\'eraire ratio yields the change-of-num\'eraire reweighting: the swap-rate law under the annuity measure, the num\'eraire-inversion involution of martingale optimal transport, and the population form of self-normalized importance sampling. We characterize exactly when it is Wasserstein-stable. Along weakly convergent inputs with uniformly integrable num\'eraires and a strictly positive limiting num\'eraire mean, convergence of the reweighted laws is equivalent to a uniform integrability condition, computed under the inputs, on one explicit family in the payoff and num\'eraire. At first order the num\'eraire cancels, the sole obstruction being payoff mass where the num\'eraire vanishes; absent such mass, prices and first moments pass to the limit under a uniformly integrable payoff alone, with no assumption on the reciprocal num\'eraire. Two-atom examples with bounded inputs attain the threshold exactly. Applications characterize a standing moment assumption in the transport literature and delimit which annuity-measure statistics survive.