Proposition: The same spatial allowance permits turnover and no turnover

Choose adjacent qubit sites $a,b$, and let

Define

Every $K_{\theta }$ has a unique ground state ${\Omega }_{\theta }$, gap one, and local interaction norms at most one. For every cut,

The states have exact PEPS representations of bond dimension at most two.

A cut separating $a,b$ has entropy

Thus the constant path $\theta (t)=0$ and a path increasing from $0$ to $\pi /2$ have the same uniform gap and spatial allowance but different entropy histories.

Proof. The pair projector has a one-dimensional kernel and gap one. All other projectors act on disjoint sites. If a cut does not separate $a,b$, its reduction is pure. Otherwise the only mixed factor has eigenvalues ${\cos }^2\theta ,{\sin }^2\theta$, with entropy at most $\log 2$. The edge $\{a,b\}$ crosses every such cut. A Schmidt expansion on that one edge gives bond dimension two. The entropy is zero at the endpoints and $\log 2$ at $\theta =\pi /4$.