Transverse tests and supported references

In coordinates $(\tau ,\nu )$ for $N$, transverse tests must include the normal origin. A term $a(\tau ){{\partial }_{\nu }}^{\alpha }{\delta }^{(8)}(\nu )$ pairs with ${\Phi }_{\lambda }$ at scale ${\lambda }^{-8-|\alpha |}$, but vanishes on every normal annular test avoiding zero.

Any smoothing-uniform transverse estimate passes to an existing distributional limit by fixing $\lambda$ and taking the limit in its test pairing. The estimate (eq:two-parameter) is not of this form, because its smoothing loss remains.

For the narrower supported sector, $S_{\mathrm{resp}}$ has five normal coordinates and

On $S_{\mathrm{resp}}\setminus N$, at most one crossed cone is singular: simultaneous vanishing of $q_1,q_2$, together with $q_R=0$, forces aligned future null segments. A crossed conormal there has a nonzero $y$-component, whereas $N^{\ast }{S}_{\mathrm{resp}}$ has zero $y$-component. The ordinary product and trace criteria hold.

Its component reference is consequently

where

This is only the $K_6+K_8+K_9$ reference. Extending it within a fixed supported normal-order class is equivalent to extending its unique tangential coefficient distributions. Polynomial normal-jet tests extract those coefficients, and their insertion gives the converse. Neither their extension nor a full-sum characteristic composition follows from the finite jet count XIVJ.