  We establish a sharp reciprocity inequality for modulus in compact metric
spaces $X$ with finite Hausdorff measure. In particular, when $X$ is also
homeomorphic to a planar rectangle, our result answers a question of K. Rajala
and M. Romney. More specifically, we obtain a sharp inequality between the
modulus of the family of curves connecting two disjoint continua $E$ and $F$ in
$X$ and the modulus of the family of surfaces of finite Hausdorff measure that
separate $E$ and $F$. The paper also develops approximation techniques, which
may be of independent interest.
