The exact common pairing

Use $M={\mathbb{R}}^4$, $W={M^4}_{(a,b,z,y)}$, and $\eta =\mathrm{diag}(1,-1,-1,-1)$. The defining kernels and complete adjoints are displayed and proved in Appendix app:adjoints. Their inventory is

There are six operational entries, since $K_9=K_{9d}+K_{9m}$.

The mathematical crossed operation uses

and

Relative derivatives are taken before pullback and raising uses $\eta$. Define

This fixes a mathematical operation. Its physical identification is an unresolved entry of the ledger.

Fix $\zeta \in {C_c}^{\infty }(W)$, a relative cutoff $\chi =1$ near $\pi (\mathrm{supp}\zeta )$, and one compact profile $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ with $\int \rho =1$. Put

The differentiated profile is

It satisfies

These identities follow by direct differentiation and integration by parts.

For a two-line row define its smooth ambient amplitude by

The inverse exists for these independent seed charts. For response rows, retain the smooth restricted amplitude

Thus the derivatives have already been applied before any geometric restriction below.

Let ${\mu }_e$ denote the scalar ambient seed measure:

These expressions mean the independent tensor-product measures of Appendix app:adjoints, not products formed after coincident-variable restriction. Write ${\mu }_0$ for the response measure