Proposition: Characteristic ambiguity at codimension four

Let $N$ be a smooth codimension-four submanifold and choose local coordinates $(s,r)$ with $N=\{r=0\}$, $r\in {\mathbb{R}}^4$. Suppose an extension $t$ of a fixed $t^{\circ }$ has transverse scaling degree at most $d$, where $d\ge 4$.

For every nonzero smooth compactly supported $a(s)$,

has transverse scaling degree four and wavefront set contained in $N^{\ast }N$. Hence $t+A$:

1. agrees with $t^{\circ }$ off $N$;

2. has transverse degree at most $d$;

3. preserves any wavefront bound for $t$ that already allows all of $N^{\ast }N$.

If this characteristic neighborhood is disjoint from the permitted interaction collision supports, the nonzero difference $A$ is not a member of ${\mathfrak{C}}_{\mathrm{part}}$.

Proof. Normal dilation gives ${\delta }^{(4)}(\rho r)={\rho }^{-4}{\delta }^{(4)}(r)$. The smooth tangential coefficient adds no wavefront covector. Thus the stated degree and conormal bound hold.

The distribution $A$ vanishes off $N$. Scaling degree and wavefront set of a sum are bounded by the corresponding maximum and union, proving (i)–(iii).

Every permitted collision change vanishes in the specified collision-free neighborhood. The nonzero $A$ does not, so it cannot belong to that class.