The adaptive coupling upper bound for the Wasserstein separation: microscopic flips against the macroscopic symmetry cost

For the extreme broken-phase KMS states of a localized Bost-Connes system, separated by the order parameter chi(g) prod_j epsilon_j(x_j), the only known upper bound on the Connes-Rieffel distance W_1 was the trivial group-flip cost w_0. This paper constructs an x-adaptive coupling and proves the closed-form bound W_1 <= w_0 Pi_infinity + sum_k w_{j_k} tau_{j_k} prod_{l