Lemma: Disconnected components and zero boundary

For (eq:H), finite-range terms lie in individual connected components. If the full ground state is unique, it is the tensor product of the unique component ground states. In particular, if ${\partial }_{\Lambda }A=\varnothing$, then $S_{\Omega }(A)=0$.

Proof. A finite-diameter support cannot meet two distinct components. Hence $H$ is a tensor sum of component Hamiltonians. Ground energies add and ground-space dimensions multiply. Uniqueness forces each component ground space to be one-dimensional. A zero-boundary set is a union of components, whose joint state is pure and uncorrelated with the remaining components.