Proposition: Pure-trace equality does not decide the stated wavefront condition

There is a compact distributional three-slot kernel, invisible to all pure-trace pairs, which violates the stated wavefront condition. Therefore identical pure-trace data can occur for both the zero kernel and a nonadmitted kernel.

Proof. Retain $A$ from Proposition prop:trace. Choose a time $t_c$, nonzero compact spatial $a$, a compact smooth $b$ whose time support is disjoint from $t_c$, and nonzero compact smooth $c$. Define

and

Trace contractions vanish.

At a point with $t_1=t_c$, $a({\mathbf{x}}_1)\ne 0$, $b(x_2)\ne 0$, and $c(x_3)\ne 0$, the second summand vanishes in a neighborhood because $b$ is supported away from $t_c$. The first summand has wavefront covectors

with zero covectors in the other slots. For $\lambda >0$ these lie in the product of closed future causal cones; for $\lambda <0$ in the corresponding past product. The required exclusion fails.