Theorem: Characteristic-conormal projection

At $w\in N$, with direction ${\ell }_{\mathbf{n}}$, the conormals surviving $\pi$ are exactly

Consequently, for

The equality is only over $\pi (N)$.

Proof. Simultaneous translation is tangent to $N$, giving total covector balance. For a surviving covector write ${\xi }_a={\xi }_b=0,{\xi }_z=p,{\xi }_y=q$. Variation of $a$ gives $p+q=0$, variation of $\alpha$ gives $q({\ell }_{\mathbf{n}})=0$, and variation of $\mathbf{n}$ gives $q(0,\delta \mathbf{n})=0$ for $\delta \mathbf{n}\perp \mathbf{n}$. Thus $q=c(1,-\mathbf{n})=c{{\ell }_{\mathbf{n}}}^{♭}$ and $p=-q$.

Conversely these covectors annihilate every tangent variation. They are precisely the residual null-cone conormals because $y-z=\alpha {\ell }_{\mathbf{n}}$. Locally the permitted $\beta$-values are open when a retained point exists, so $\pi (N)$ is locally an open part of $\mathcal{H}$.