Theorem: Failure of the automatic two-factor pullback step

At every retained point $w\in N$ where the diagnostic weights are nonzero, the two defined factors in (eq:UV) admit opposite nonzero wavefront covectors:

and

Therefore the ordinary two-factor product criterion does not define $UV$ at these configurations.

The individually assigned relations and symbols do not justify this final product step.

Proof. Choose the individual normal frequencies

and choose zero attachment covector.

The factor symbols in Proposition prop:factorsymbol are elliptic at these conormal selections. Hence the selected directions are actual individual wavefront directions, not merely members of an upper bound.

The copied positive relation gives their zero sum. Thus the wavefront sets of $U$ and $V$ contain opposite nonzero covectors at the same point. The sufficient product hypothesis fails Hormander (Theorem 8.2.10).