Entropy Moduli and Support-Sensitive BKM Coercivity for Rank-Deficient Non-Commutative Markov Semigroups

We study entropy--coherence relations near rank-deficient support boundaries in finite-dimensional quantum systems. For block-diagonal reference states, we establish support-sensitive coercivity estimates showing that the entropy cost of cross-boundary coherence acquires a logarithmic enhancement as the population scale approaches the support boundary. Combined with finite-time entropy bounds, these estimates yield conditional entropy--activation bounds with a logarithmic correction factor of order $e^{-\alpha t}(1+\alpha t)^{-1/2}$ in coherence-dominant regimes. The analysis proceeds through pinching reductions and effective $2\times2$ Bogoliubov--Kubo--Mori (BKM) estimates adapted to the coherence--population structure. We further apply the framework to Davies semigroups under additional secular decoupling and population-rate assumptions. The resulting statements provide conditional certification bounds near rank-deficient stationary states, rather than general mixing-time or convergence-rate estimates.