Witness notation and the exact quantifier

Fix $|\mathbf{n}|=1$ and use

The source and response are those of revised XVI-B and XVI-D:

Here

and

For reference, define the independently checked witness slot

Equation (eq:witness-chain) identifies this scalar functional with the witness restriction of ${\mathcal{O}}_{\mathbb{C}}$.

The equality holds for arbitrary assignments $\lambda \mapsto {\varphi }_{\lambda }$ of real smooth scalars, at each frozen frequency. No regularity in $\lambda$ is needed, and no decay is implied by the equality alone.

There is no need for another comparison of coordinate prescriptions to prove this witness equality: the definitions match, and the supplied compatibility theorem supplies the operator comparison on the witness slot. This closes the former open item in XVI-H-v3.

The result does not:

• insert a pair as one metric argument of ${\mathcal{C}}_{3,\mathrm{D}}$;

• construct a metric $\gamma$ with $u_{\mathrm{D}}[\gamma ]=R_{\lambda }$;

• establish a general extension of a retarded operator to arbitrary distributions.

The diagonal insertion and its polarization remain differently defined objects with different input structures.