Relative entropy in finite Markov-chain averaging with zero initial block masses

We consider a fixed finite continuous-time Markov chain with generator L^epsilon = epsilon^(-1) Q + G, where Q is block diagonal with irreducible blocks. The averaged chain uses the stationary law of each fast block and starts with the same block masses as the full chain. On every fixed finite interval [0,T], the relative entropy of the true block marginal with respect to the averaged law is at most C_T min{t,epsilon}, uniformly over all initial probability laws, including zero initial block masses and laws depending arbitrarily on epsilon. A shortest-path comparison bounds the likelihood ratio uniformly and converts the averaging error into an entropy estimate at zero initial masses. For two blocks we obtain the sharper bound C_T min{t,epsilon^2/t} and an explicit lower bound on newly created block mass. A fixed three-state example attains order epsilon at t = epsilon. We also treat the effective chain defined by the conditional stationary law of the full generator, keeping it distinct from the averaged chain.