An explicit choice of local collision coordinates

Let a graph have vertex-coordinate space $X_{\Gamma }$ and fix $I=\{i_0,i_1,\dots ,i_k\}$. In a coordinate neighborhood choose

Let $s_I$ consist of $x_{i_0}$ and the coordinates of vertices outside $I$. Then

The inverse recovers $x_{i_j}=x_{i_0}+r_{I,j}$ and leaves the other vertices unchanged. This is an explicit coordinate completion of role (i), not a new collision prescription.

The codimension is $4k=4(|I|-1)$ in the graph coordinate space. Coordinate-density conversions, if needed for a different local chart, are smooth coefficient factors. They do not alter which collision is being represented and do not identify this graph codimension with the codimension-eight locus $N$ in $W$.