Theoretical analysis reveals degeneracies in twisted transport metrics on Bost-Connes systems, indicating that local functional inequalities fail to detect the system's phase transition.
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.
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Ruqing Chen (2026) studied this question.
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