Theorem: Injective degree-four presentations

The maps $p_L:E_L\to {\mathcal{L}}_4$ and $p_R:E_R\to {\mathcal{R}}_4$ are algebraic isomorphisms onto their images. Thus

Proof. Suppose

Differentiate once in a scalar direction $\phi$ and three times in metric directions $k,a,b$, at zero, where $k_{00}=0$, $a_{00}=u$, $b_{00}=z$. Only the assignments placing $k$ in the response slot survive. The result is

For

Lemma lem:restricted gives $f_{u,z}=0$. Any compact $y$-test $d$ is such a product: take $u=d$ and $z=1$ near $\mathrm{supp}d$. Every $y$-test contraction of $v$ is therefore zero, so $v=0$. Faithfulness of the algebraic test embedding finishes the left case. The right case is the same classical product with exchanged test factors. Proposition prop:prelations gives the last equivalence.