Example: An area allowance alone does not imply spatial information decay

On a connected square lattice, the GHZ vector has entropy $\log 2$ for every nonempty proper subset, so it satisfies an area allowance with coefficient $\log 2$. Two separated nonempty subsystems have mutual information $\log 2$ whenever their union has nonempty complement. This example is not asserted to be a unique uniformly gapped local ground state. It shows why the collar estimates cannot be recovered from an entropy allowance alone.