Lemma: Component conversion audit

In the basis (eq:basis), define

For the ordered labels $(00,01,02,03,11,12,13,22,23,33)$,

If ${\mathcal{E}}^{\mu \nu }$ is a contravariant Euler derivative, its coordinate Euler components and propagated covariant components are related by

For arbitrary symmetric $h,k$,

No additional diagonal/off-diagonal weights occur in a contraction of a coefficient dual to $h_I$ with $h_I$.

Proof. The basis tensors have disjoint component supports. A diagonal contraction has one term; an off-diagonal contraction has two terms and the corresponding metric sign. This gives the displayed diagonal matrix.

For $I=00$, $E_{00}={\mathcal{E}}^{00}$ and $J^{00}={\mathcal{E}}^{00}$. For $I=01$,

These agree with the corresponding entries of ${\mathsf{G}}^{-1}$.

The general statement follows from the completeness identity

It also proves the full-index pairing formula after substituting the component expansions. Therefore an extra off-diagonal factor in an already dualized coefficient pairing would count the same conversion a second time.