Theorem: Pure-trace witness non-determination

The complete family eq:witnessmap, for all $\lambda >0$, unit $\mathbf{n}$, and scalar compact smooth tests, does not determine:

1. the compact three-slot kernel;

2. its membership in ${\mathcal{K}}_{\mathrm{mc}}$;

3. its class modulo ${\mathcal{I}}_{\mathrm{sol}}$.

Proof. Choose a nonzero constant contravariant symmetric tensor $A$ such that $A^{\mu \nu }{\eta }_{\mu \nu }=0$, for example $A^{12}=A^{21}=1$ with other components zero. Fix $y_0\in M$. Choose a nonzero nonnegative $v\in {C_c}^{\infty }(M)$ supported where ${\varphi }_m(t,\mathbf{x})=\cos (mt)>0$; for $m=0$, take ${\varphi }_0=1$. Define

This is a nonzero symmetric compact kernel. Since $A:\eta =0$, every pure-trace witness contraction of $w$ vanishes. Thus $w$ and zero have identical witness data, proving (i).

At a point where $v(x_3)\ne 0$, the wavefront set of the two point deltas contains two nonzero future causal covectors in the tensor slots and zero in the scalar slot. Consequently $w\notin {\mathcal{K}}_{\mathrm{mc}}$, while $0\in {\mathcal{K}}_{\mathrm{mc}}$, proving (ii).

Choose a constant tensor $H$ with $A:H\ne 0$. Then $\square H=0$ and $P_m{\varphi }_m=0$, but

Hence $w\notin {\mathcal{I}}_{\mathrm{sol}}$. Its class differs from the zero class, proving (iii).