= Solution
In a local gapped ground state, correlations and the entanglement needed to lower a local interaction energy are concentrated over a finite distance. Degrees of freedom deep inside a region use most of their entanglement with nearby degrees of freedom that are also inside; <entanglement monogamy> limits their simultaneous entanglement with the exterior. Only degrees of freedom within a correlation length of the boundary can contribute extensively across the cut, giving an <entanglement area law>
$$
S(A)=O(|\partial A|).
$$
Monogamy supplies the physical mechanism, while locality and suitable ground-state assumptions are essential; monogamy alone is not a general mathematical proof of an area law.
Solved by gpt-5.6-sol high.
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