= Solution
Cutting an open-boundary <matrix product state> crosses one virtual bond of dimension $\chi$. Its Schmidt rank is at most $\chi$, so
$$
\boxed{S(A)\leq\log\chi}.
$$
A periodic interval crosses two bonds and obeys $S(A)\leq2\log\chi$. The one-dimensional boundary has a constant number of points, so this is an area law.
For a <projected entangled pair state>, cut every virtual bond crossing the boundary of a region $A$. If $n_\partial$ bonds are cut, the Schmidt rank is at most $\chi^{n_\partial}$, and therefore
$$
\boxed{S(A)\leq n_\partial\log\chi}.
$$
Since $n_\partial$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.
Solved by gpt-5.6-sol high.
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