Theorem: Non-clean elementary product phase

At the retained characteristic configuration, the seven auxiliary critical equations of (eq:phase) have differential rank six. Their critical set has codimension eight in the twenty-three-dimensional base/frequency space.

Thus:

1. the candidate is not nondegenerate;

2. it is not clean under the stated tangent-space criterion;

3. on the ray

   $$t_R=\frac{c}{2\beta },t_1=\frac{c}{2\alpha },t_2=\frac{c}{2(\beta -\alpha )},\kappa =0,c>0,$$

   its entire differential vanishes.

It cannot be used as the ordinary common phase defining a principal symbol of (eq:target).

Proof. The three cone differentials have copied rank two. The four components of $\mathrm{d}(a-z)$ are independent of them: looking at their $z$ components forces every coefficient in a proposed linear relation to vanish. The total rank is therefore six.

By Proposition prop:lifts and Lemma lem:zerolocus, the critical set has dimension

inside a space of dimension $16+7=23$. Its codimension is eight.

The kernel of the differential of the critical equations has dimension $23-6=17$, whereas the tangent space of $C_{\Phi }$ has dimension fifteen. They are unequal. Independence and the clean tangent-space condition both fail.

On the displayed frequency ray, the base differential is

by the copied positive relation. The auxiliary differential also vanishes on $C_{\Phi }$. Thus $\mathrm{d}\Phi =0$ there even though the frequency variables are nonzero.