Theorem: Injectivity of the degree-four presentations

The maps

are linear isomorphisms onto their actual-functional images. In particular,

Proof. Suppose $p_L(v)=0$. Choose metric directions $k,a,b$ with

where $u,z\in {C_c}^{\infty }(M_{+})$; the other components of $a,b$ may be zero. Differentiate once in a scalar direction $\phi$ and three times in these metric directions at the zero configuration.

Using the unsymmetrized functional expression,

the only surviving terms place $k$ in the $Lh$-slot. Consequently the derivative is exactly

The factor $2$ is the surviving permutation count after $k_{00}=0$; it is consistent with the $3!$ recovery of the fully symmetric coefficient.

Define

It is a smooth compact test. Since (eq:isolation) vanishes for all $k,\phi$, Lemma lem:restricted gives $f_{u,z}=0$ for every $u,z$.

For any $d\in {C_c}^{\infty }(M_{+})$, take $u=d$ and choose $z$ equal to one near $\mathrm{supp}d$. Then

for every $x$. Thus every $y$-test contraction of $v$ vanishes, so $v=0$ as a joint smooth function and hence as an algebraic tensor.

The right map is the same classical product after $v_R\mapsto \tilde{v}$. Therefore its kernel is also zero. Surjectivity onto the named images is their definition.