Quantum optimal transport on Bost-Connes systems: the twist is harmful, the generator has a sign error, and the transition is invisible to local inequalities

Quantum optimal transport is set up on localized Bost-Connes systems. The GKLS generator must be taken in symmetrized form, the proposed twisted transport metric is shown to be harmful (cross-prime cancellation makes the twisted couplings degenerate), and tensorisation reduces modified log-Sobolev constants to an infimum over primes controlled by the single-prime gap, with the weight dichotomy showing that the phase transition is invisible to every local functional inequality.