Ambient realization and scalar product data

We use the surrounding Minkowski space $M={\mathbb{R}}^4$ to define distributional kernels. The observation and switch remain on the slab. The ramp is smooth on $M$ and zero for $t\le 0$. All insertion kernels below are compact on $M^n$.

Smooth free slab solutions extend to global free solutions by their Cauchy data. Thus restricting these compact kernel functionals to global solutions gives the same interpretation on ${\mathcal{C}}_{\mathrm{sol}}$ as the original slab prescription. No boundary test is silently replaced by an interior test.

For $\mu \ge 0$ set

and define the scalar bidistribution

It is interpreted distributionally, including the massless case. The spatial Fourier formula for ${\delta }_{\mu }$ gives

To match the proposed block notation put

These are bisolutions with one common oriented Hadamard wavefront relation. Their normalization follows directly from (eq:W); it is not copied from another metric action.

We use these distributions as product data. No functional $\omega$ on the resulting algebra is selected, and no expectation value is computed.