The convention

The joint-map statements of XIV-I and XIV-J name a residual species with a metric slot at $z$ and a scalar slot at $y$ II3, Jmapclass; they do not display the bundle. The following convention is introduced here and offered as the frozen choice.

1. Fiber. $E_{\mathrm{res}}:={\mathrm{pr}}_z^{\ast }\left({\mathrm{Sym}}^{2}TM\right)\otimes {\mathrm{pr}}_y^{\ast }1$, where ${\mathrm{pr}}_z,{\mathrm{pr}}_y:W\to M$ are the coordinate projections and $1$ is the trivial line (the scalar slot at $y$). The fiber at $w=(a,b,z,y)$ is ${\mathrm{Sym}}^2{T}_{z}M$, of dimension $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$; $E_{\mathrm{res}}$ is a trivial bundle of rank ten over $W$.

2. Density. The density bundle $|\Lambda {|}_W$ is trivialized by the coordinate density $|{\mathrm{d}}^{16}w|$ of $(a,b,z,y)$.

3. Tests. ${\Gamma }_c:={C^{\infty }}_c(W;{E_{\mathrm{res}}}^{\ast }\otimes |\Lambda {|}_W)\cong {C^{\infty }}_c(W)\otimes {\mathrm{Sym}}^2({\mathbb{R}}^4{)}^{\ast }$; an element is a symmetric array $\Phi =({\Phi }_{ab}{)}_{a,b=0}^3$ of test functions on $W$.

4. Pairing. ${\mathcal{D}}^{\prime }(W;E_{\mathrm{res}}):={\Gamma }_c^{\prime }$, with $⟨u,\Phi ⟩={{\sum }_{a,b=0}}^3⟨u^{ab},{\Phi }_{ab}⟩$ for $u=(u^{ab})$ a symmetric array of scalar distributions; the fiber pairing ${\mathrm{Sym}}^{2}V\times {\mathrm{Sym}}^2{V}^{\ast }\to \mathbb{C}$, $(u,\Phi )\mapsto u^{ab}{\Phi }_{ab}$, is the restriction of the duality $V^{\otimes 2}\times (V^{\ast }{)}^{\otimes 2}\to \mathbb{C}$ and is perfect.