Proposition: Exact positive-cutoff distribution and placement

The field regularization and the mixed-kernel convolution in (sc:placement) define the same insertion. Their numerical coefficient is

For every $\epsilon >0$, this is a smooth function on $X$, and

Its pairing on real product tests is exactly (sc:spectral).

Proof. and kernel convolution. For $\check{\rho }(z)=\rho (-z)$, transposition of convolution gives

Since $\rho$ is even, this is the kernel convolution displayed in (sc:placement). The identity holds on arbitrary smooth configurations: the original mixed kernel has compact support, and the convolution kernels are compactly supported.

-wave multiplier identities. The coefficients of $C$ are constant, so

for every smooth $h$. For any constant tensor $v$,

This applies in particular to the de Donder polarizations in (sc:covariance). Applying the fixed operator ${\mathsf{E}}_{0}C$ to this equality retains the scalar multiplier $\vartheta (\epsilon P)$. This proves the displayed formula for $K_{\epsilon }$, without commuting $S_{\epsilon }$ through ${\mathsf{E}}_0$.

For the scalar leg, Euclidean radiality gives

The two scalar insertion slots therefore supply ${M_{\epsilon }}^2$. The derivative factors are

multiplying the scalar spectral density $1/(2{\omega }_m(q))$ gives (eq:Deps). The two insertion signs also multiply to $+1$. The middle observable is untouched.

and the product-test identity. For fixed positive $\epsilon$, every coordinate derivative of the momentum integrands in $K_{\epsilon }$ and $D_{\epsilon }$ is integrable, locally uniformly in the observations. At zero momentum the only graviton denominators are $P^{-1}$ and $P^{-2}$, locally integrable in three dimensions. The scalar density is locally integrable also for $m=0$. At infinity the profiles provide arbitrary polynomial decay.

Thus (eq:Texact) is a product of smooth functions. Its compact-test bound follows by a supremum on a compact neighborhood of $K$. Integrating against $f\otimes g\otimes f^{\prime }$ and using (eq:Deps) yields (sc:spectral).