Witness contractions do not determine the kernel

Let $E_h=S^2{T}^{\ast }M$. Let $\mathcal{K}$ consist of compactly supported distributional kernels on $M^3$ with two dual symmetric-tensor slots and one scalar slot, symmetric under exchange of the tensor slots. Fixed trivializations make the component description unambiguous. Compact support allows pairing with arbitrary smooth configurations.

Let ${\tilde{V}}_{\pm }$ be the future or past causal covector cone with zero adjoined. Define

This is a wavefront admission condition, not a gauge identity.

For $|\mathbf{n}|=1$ and $\lambda >0$, put

where $k_a\cdot x$ denotes the covector pairing. The covectors are null, so $\square {{\gamma }_a}^{\lambda }=0$. The scalar partial contraction is

The pairing is defined by compact support. It does not assert existence of a quantum contraction.

Define the free-shell null space

It is a vector subspace. Its quotient class is $[u{]}_{\mathrm{sol}}=u+{\mathcal{I}}_{\mathrm{sol}}$. These free equations are not the coupled Einstein equations.