Fermi–type field $\phi$ with anticommutator $$\label{counterexample}
\phi(x)\phi(y) + \phi(y)\phi(x) = (W(x,y) + W(y,x)) \cdot 1 \, ,$$ where $W(x,y)$ is taken to be any ${\mbox{SO$_0(2,n\!-\!1)$}}$–invariant two-point function satisfying the spectrum condition. (Specific examples are provided by two-point functions given in [@F2 eq. (4.9)]). This anticommutator does not vanish when $x$ and $y$ are in complementary wedges. The field $\phi$ generates a CAR-algebra with a quasifree state fixed by the two-point function $$\omega( \phi(x) \phi(y)) = W(x,y) \, .$$ Proceeding to the GNS representation with cyclic vector $\Omega$, we conclude from [@F2; @BEM] that this example satisfies all of our standing assumptions. Since $W(x,y)$ is symmetric when $x,y$ are in complementary wedges, the weak locality is explicit.

As explained in [@AdS], if (\[locality\]) holds there are grounds to expect that the theory generically cannot have interaction. Theories which are weakly local but not strongly local escape that argument. In the following sections we shall therefore continue to explore consequences of Standing Assumptions (i)–(iv) when (\[locality\]) does not hold, beginning with the nature of the independence of the algebras $\As(W_1)$, $\As(W_2)$ associated with suitable spacelike separated wedges. But, first, we make some further observations.

Assumptions (i)–(iv) entail that each wedge algebra $\As(W)$ 