Proposition: Relation, equality diagonals, and density

The reflected relation has dimension eight and codimension four in $X$. It is not an observation equality diagonal. Moreover,

and

where that intersection is present.

Near a point of $\mathcal{R}$, set

Then

In particular, no $\delta (t-t^{\prime })$ occurs.

Proof. The parameter space has dimension $3+2+3=8$. The differences

prove all diagonal statements. Equality $x=x^{\prime }$ is exactly the one additional relation $t=t^{\prime }$, when restricted to $\mathcal{R}$.

For the density, the spatial change to $(\mathbf{y},\mathbf{x}-\mathbf{y},\mathbf{w})$, at fixed times, has determinant of absolute value one. Write $\mathbf{x}-\mathbf{y}=(a+\sigma )\mathbf{n}$, producing the polar factor $(a+\sigma {)}^2$. The replacement of $\mathbf{y}$ by $\mathbf{z}$ is triangular with unit determinant. The time block is unchanged. The inverse coordinate formulae are smooth near the relation. Restricting the density to $\sigma =\mathbf{w}=0$ and comparing with $\mathrm{d}\nu$ gives the coefficient $c_0{a}^{-2}$.