Proposition: The retained stratum is not a point puncture

On the stated nondegenerate patch, the stratum in (eq:N) has dimension eight and codimension eight in $W$.

Proof. The parameters are $a\in {\mathbb{R}}^4$, the two scalars $\alpha ,\beta$, and $\mathbf{n}\in S^2$, giving dimension $4+2+2=8$. The displayed parametrization has a smooth inverse on its image:

The positivity of $\alpha$ makes the inverse smooth. Thus the parametrization is an embedding on the retained patch. Since $\dim W=16$, the codimension is eight.