Proposition: A conditional nonuniqueness diagnostic in W

In a chart $(\tau ,\eta ,n)$ as above, a nonzero smooth compactly supported $a(\tau )$ defines

It is supported in $N_U$ and has wavefront contained in $N_U^{\ast }U\setminus 0$.

It is not a member of the inherited interaction collision-contact class on this retained separated patch.

Accordingly, agreement off $N_U$ and a wavefront allowance containing the full ambient conormal would not, by themselves, select a unique extension. This statement is conditional on such an allowance and does not assert an actual degree of the target.

Proof. The delta in the eight normal coordinates is a defined distribution. Its wavefront is conormal to their common zero set, and multiplication by the smooth tangential coefficient preserves that upper bound.

Every permitted interaction collision contact restricts to zero on the retained patch by the copied rigidity statement. The nonzero distribution (eq:AN) does not, so it is not such a contact.

Adding (eq:AN) to an existing extension would preserve agreement off $N_U$ and any wavefront upper bound that already permits its conormal. No extension is being constructed or modified here.