 Solution

In a local gapped ground state, correlations and the entanglement that lower local interaction energies are concentrated within a finite correlation length. A degree of freedom deep inside a region uses most of its entanglement with nearby degrees of freedom that are also inside. Entanglement monogamy limits its simultaneous entanglement with the exterior, so only degrees of freedom within a correlation-length layer of the boundary can contribute extensively across the cut. Their number scales as the boundary area, suggesting the entanglement area lawThis is a physical argument rather than a proof from monogamy alone; locality and suitable ground-state assumptions are essential.