Assumption: Environmental moment certificate

The chosen reduced dynamics preserves normality and finite number moments. The function $n_{ϵ}(s)=\mathrm{Tr}({\rho }_{ϵ,s}N)$ is absolutely continuous and satisfies, almost everywhere,

where $\alpha ,\beta$ are integrable and $\beta \ge 0$. These statements must be justified from the chosen dynamics; they are not consequences of static concentration.