Proposition: Characteristic and attachment codimensions

Locally at the retained point,

The intersection defining ${\tilde{\mathcal{N}}}_{\mathrm{att}}$ is transverse.

Proof. The map $P$ is a submersion, so the copied codimension-four stratum lifts to a codimension-four submanifold with $z$ free.

The equation $a-z=0$ has four independent components. Every conormal of ${\overline{\mathcal{N}}}_{\mathrm{att}}$ has zero $z$ component, whereas a conormal to ${\mathcal{D}}_{\mathrm{obs}}$ has opposite $a$ and $z$ components. A covector belonging to both conormal spaces must consequently have its $z$, then its $a$, component zero, and hence is zero. This proves transversality and the codimension-eight conclusion.