Proposition: Individual conormal symbols at the diagnostic directions

On ${\Lambda }_U$, near the selected retarded direction, $U$ has normal amplitude order one and hence conormal order

Its leading normal amplitude is a nonzero conventional factor times

On ${\Lambda }_V^{+}$, in the stated positive comparable-frequency sector, $V$ has normal amplitude order two and hence conormal order

Its leading normal amplitude is a nonzero conventional factor times

Both individual symbols are nonzero at the copied characteristic selections. They are not a joint symbol of their product.

Proof. The normal amplitude of a delta distribution has order zero. A derivative acting on its normal singularity multiplies its leading amplitude by $i{\theta }_R{\partial }_{a^0}{q}_R$. Multiplication by the attachment delta adds four independent normal variables with order-zero amplitude. This gives (eq:Usymbol) and its order.

For $V$, the two normal defining functions are independent. In their separate adapted normal coordinates the simple-pole amplitudes have order zero on their selected half-rays. The two temporal derivatives of the second line give the leading degree-two factor in (eq:Vsymbol). Derivatives of amplitudes and the massive lower normal terms have lower order in this sector. The usual individual conormal symbol calculus applies because these normal coordinates are regular Wunsch (Section 8).

On $N$, with $\delta =\beta -\alpha >0$,

The selected normal frequencies are nonzero, as are $c_R,c_1,c_2$. Thus neither leading amplitude vanishes there.