  We show that in two spatial dimensions, when a quantum state has entanglement
entropy obeying a strict area law, meaning $S(A)=\alpha |\partial A| - \gamma$
for constants $\alpha, \gamma$ independent of lattice region $A$, then it
admits a commuting parent Hamiltonian. More generally, we prove that the
entanglement bootstrap axioms in 2D imply the existence of a commuting, local
parent Hamiltonian with a stable spectral gap. We also extend our proof to
states that describe gapped domain walls. Physically, these results imply that
the states studied in the entanglement bootstrap program correspond to ground
states of some local Hamiltonian, describing a stable phase of matter. Our
result also suggests that systems with chiral gapless edge modes cannot obey a
strict area law provided they have finite local Hilbert space.
