C3: scaling degrees and their exact scope

For a distribution in normal coordinates $r\in {\mathbb{R}}^q$, its scaling degree is the infimum of $s$ such that ${\rho }^{s}t(\rho r)\to 0$ distributionally as $\rho \downarrow 0$. For ${\Delta }_I$, the transverse codimension is

The tangential variables, including an uncollapsed observation point, remain variables of the distribution.

The extension theorem gives uniqueness below codimension and a finite derivative-of-delta ambiguity at or above codimension, provided the relevant distribution and scaling estimates exist. The submanifold version additionally requires tangential and microlocal control BF,Dang (Sections 5–6).