Summable-memory coupling ($\Vert q(\cdot \mid \eta )-q(\cdot \mid {\eta }^{\prime }){\Vert }_{\mathrm{TV}}\le b_r,{\sum }_r{b}_r<\infty$)

Suppose a finite-alphabet transition kernel is uniformly positive and histories agreeing in their most recent $r$ symbols have total variation distance at most $b_r$. Choose decreasing bounds with $b_r<1$ and summable tail. A maximal coupling can be dominated by a match-length chain that increments with probability $1-b_r$ and resets with probability $b_r$. Its chance of never resetting is ${\prod }_{r\ge 0}(1-b_r)>0$. Successive failed attempts therefore end almost surely, leaving a final infinite run of matches. The dominating chain does not depend on the histories, so future-block laws forget the remote past uniformly. This proves uniqueness of a two-sided law with the prescribed transition kernel without assuming stationarity first.