State-Level Kernels of Downward-Transition GKSL Dissipators: A Conditional Application to C-Parity Models

We characterize the state-level kernel of a restricted class of Gorini–Kossakowski–Sudarshan–Lindblad dissipators with a strict downward-transition structure. Let a finite-dimensional effective system admit an orthogonal decomposition into lower- and upper-energy sectors. We assume that every active jump operator maps the upper sector into the lower sector and that the sum of the corresponding rate-weighted transition operators is uniformly positive on the upper sector. Under these hypotheses, we prove that the density operators annihilated by the dissipator are exactly those supported on the lower sector. The same assumptions imply exponential decay of the upper-sector population. The result gives a direct and quantitative specialization of the broader theory of quantum dynamical semigroups, invariant subspaces, stationary states, and dissipative stabilization. For this triangular class, two directly verifiable conditions on the jump operators yield both an exact state-level kernel characterization and an explicit exponential upper-sector population bound. We finally apply the theorem conditionally to a C-parity carrier model in which the C-even and C-odd representation sectors are assigned to the lower and upper sectors, respectively.