Proposition: Exact degree-four relation equations

With the coefficient convention

one has

For $v_R(y,x^{\prime })$, let $\tilde{v}(x^{\prime },y)=v_R(y,x^{\prime })$. Then

Proof. Insert the kernel of $L$ in (eq:generators), multiply by $W_g$, and integrate the scalar observation derivative by parts. The original insertion sign changes the coefficient to the positive derivative in (eq:pkernel).

Differentiating once in the scalar direction and three times in independent metric directions recovers the symmetric coefficient kernel, multiplied by $3!$. Polarization and density of product tests prove uniqueness. The right formula uses only pointwise commutativity of the original classical factors.