Theorem: Exact complete-contraction representation

The compact distribution ${\Omega }_{\Phi }$ is well defined, and

All seven rows and all three vertex types are retained.

If $\chi C$ has order at most $p$, then locally

on a fixed compact neighborhood of its support.

Every equality of actual internal distributions is respected. In particular, a reconstruction-null presentation gives ${\Omega }_{\Phi }=0$ for every test.

Proof. Transposing (eq:vertextest) for a smooth symmetric array $H$ gives

The kinetic test sign $-i$ becomes $+i$ on transposition. The trace-kinetic sign $+i/2$ becomes $-i/2$. The mass sign remains $-i{m}^2/2$. Symmetry of $H$ gives the first-line symmetrization in (eq:Lambda).

Insert $H={\rho }_{\delta }\ast (\chi C)$. The resulting tensor pairing is

The factors are compact, so a compact cutoff can be inserted near their product support. Pushforward by $\mathfrak{d}$ gives (eq:exacttranslation).

The internal order-four estimate gives order at most five for $\Lambda [\Phi ]$, with a factor $\Vert \Phi {\Vert }_5$. Combining at most five $x$-derivatives and $p$$v$-derivatives on $\psi (x-v)$ gives (eq:Omegaorder).

Every operation starts from the actual $K$ and is linear. A zero reconstructed distribution therefore remains zero.