Theorem: Point-reduced extension freedom

For the finitely many numerical products occurring in the graph, extensions

exist without increasing their scaling degree.

For a fixed pair with $n=|\mu |+|\nu |$, two such extensions differ by

In particular, all building blocks in the source-decorated sector have ambiguity order at most four.

Replacing the numerical products in (eq:Vblock)–(eq:U) by these extensions defines distributions ${\bar{\mathcal{V}}}_{IJ}^{\alpha \beta }[h]$ and ${\bar{U}}_A[h]$ on all $M^2$. No coefficients $a_{\mu \nu ;\gamma }$ are chosen.

Proof. The point-extension theorems for finite scaling degree apply in four relative dimensions BF (Theorems 5.2–5.3). The difference of two extensions is supported at $r=0$. A point-supported distribution is a finite sum of derivatives of $\delta$, and

The scaling bound $4+n$ therefore permits only $|\gamma |\le n$.

Pull back each extension under the submersion $(y,w)\mapsto y-w$, multiply by the displayed smooth coefficients, and take the finite component sums. These operations define the claimed global distributions.