Proposition: Kernel matching on the specified witness pair

For each $\lambda >0$, partial evaluation of the compact kernel on the two witness tensors defines a compact scalar distribution $q_{\lambda }$ by

It satisfies

with the fixed volume convention. In particular,

Proof. The witness tensors are smooth, although not compactly supported. Compactness of $t_{\mathrm{D}}$ makes (eq:q-partial) meaningful.

More explicitly, choose smooth compact cutoffs ${\rho }_1,{\rho }_2$ equal to one near the first and second support projections of $t_{\mathrm{D}}$. Then

This is independent of those cutoffs and continuous in $\varphi$. Equivalently, multiply ${t_{\mathrm{D}}}^{\mathbb{C}}$ by the smooth witness coefficients, contract its tensor indices, and push forward to the scalar slot. The pushforward is proper because the kernel is compact.

By Theorem thm:trep and algebraic complexification, the left side of (eq:t-to-j) equals ${\mathcal{O}}_{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda };\varphi )$. Theorem thm:Opol(6) identifies this with $I_{\lambda }[\varphi ]$. Equation (eq:j-pair) proves equality for every real smooth scalar.

For a complex compact test, apply that equality separately to its real and imaginary parts. Complex linearity proves (eq:q-equals-j) as a distributional equality.