The locked residue

The map $\Psi$ is a proper embedding over compact subsets of $X$. Its inverse on its image is given by the observation times and

On the preimage of a compact in $X$, the time coordinates remain bounded and positive, the spatial coordinate $\mathbf{z}$ is bounded, and $\mathbf{n}\in S^2$. The preimage is closed in this bounded parameter region and hence compact.

It follows directly that $r$ is a positive order-zero distribution, with support exactly $\mathrm{Rel}=\Psi (Z)$. Its product-test pairing is (sc:residue). The separated-support nonzero witness is already established in VII-C, lem:separated.

Relative to the ambient density (eq:density),

Indeed, multiplying this expression by $(a+\sigma {)}^2$ and integrating in $\sigma ,\mathbf{w}$ gives exactly (eq:rjoint). There is no $\delta (d)$ in this measure.

If the same embedding and parameter density are written as $r_{\Sigma }=c_{\Sigma }{\Psi }_{\ast }\nu$, their normalization comparison is

This paper always subtracts (eq:rdef). The fact that the actual leading graph coefficient equals this prescribed distribution is proved in Proposition prop:ledger; it is not built into the definition of the radial family.