Operational and free-energy convergence do not determine kinetic geometry in logarithmic quantum transport

For every primitive GNS-symmetric quantum Markov generator on M_d(ℂ), d≥2, we construct reversible extensions on a countable direct sum with a common all-time diamond-norm semigroup limit and a common trace-norm Γ-limit of relative entropies. Their entropy-constrained logarithmic current actions nevertheless have a continuum of distinct Γ-limits. The additional cotangent form is χ(E−F₀(B))‖Q₀Ce_h‖², where χ=limsupₙ(bₙ/φₙ²) is determined by the auxiliary rates. The joint action–Fisher functional has instead the original core limit. For canonically embedded faithful initial data, the corresponding energy–dissipation functionals Γ-converge, and their almost minimizers converge to the core semigroup. Every recovery family attaining a strict kinetic saving along a faithful C² curve below the entropy cap has integrated Fisher cost bounded below by a positive multiple of log²(1/δ). At fixed coupling, the current action need not be lower semicontinuous and can strictly exceed the metric energy of its induced distance. A second construction has a faithful stationary limit, vanishing generator and all-time semigroup differences in diamond norm, and entropy Γ-convergence, but collapsing distances between fixed faithful states. A bound in terms of the diamond norm of the entire auxiliary generator gives a complementary obstruction.