Localized Bost-Connes systems IV: noncommutative metric geometry

The localized Bost-Connes algebra is made a compact quantum metric space. Lipschitz seminorms adapted to the prime coordinates are constructed, the tensorization of spectral gaps across primes is controlled (Lemma 6.1), the exclusivity theorem shows the tail observables lie outside every Lipschitz domain (Theorem 4.2), the Wasserstein distance W_1 between the extremal equilibrium states is squeezed between explicit weight-dependent bounds (Theorem 5.3), and its behaviour across the phase transition is determined (Theorem 6.2).