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

For every primitive GNS-symmetric quantum Markov generator on
$M_d(\mathbb C)$, $d\ge2$, we construct reversible extensions on a
countable direct sum with a common all-time diamond-norm semigroup limit
and a common trace-norm $\Gamma$-limit of relative entropies. Their
entropy-constrained logarithmic current actions nevertheless have a
continuum of distinct $\Gamma$-limits. The additional cotangent form is
$\chi(E-F_0(B))\|Q_0Ce_h\|^2$, where
$\chi=\limsup_n b_n/\varphi_n^2$ 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 $\Gamma$-converge, and their almost
minimizers converge to the core semigroup. Every recovery family attaining
a strict kinetic saving along a faithful $C^2$ curve below the entropy cap
has integrated Fisher cost bounded below by a positive multiple of
$\log^2(1/\delta)$. 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 $\Gamma$-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.