Proposition: Balanced characteristic ambiguity

Let $\iota :N\hookrightarrow W$, and choose a nonzero nonnegative smooth compact density $a$ on a small portion of $N$. Then

is nonzero, supported in $N$, and has wavefront in $N^{\ast }N\setminus 0$. Moreover

It is not a permitted collision contact on this patch.

Thus off-$N$ agreement and a balanced residual cone do not alone select an extension if its intermediate allowance contains this characteristic conormal.

Proof. In local coordinates $A_N$ is a smooth compact tangential density times an eight-normal-variable delta. It is nonzero and conormal. A residual nonnegative test equal to one on the projected support gives the positive integral of $a$, proving nonzero pushforward. The wavefront bound follows from Theorem thm:projection and (eq:pushforward). It does not vanish on the patch, unlike every allowed collision contact.