Lemma: Completeness of the branch ledger

Equations (eq:allmixed) and (eq:allscalar) are the ordinary second derivatives of all eleven inherited branches. No tensor component, source coefficient, filled-slot leg, or scalar-slot bijection has been discarded.

Proof. For $B_1,\dots ,B_7$, vary each of the two metric occurrences in turn, and replace the single scalar by $\psi$. This gives ${\mathcal{M}}_1,\dots ,{\mathcal{M}}_7$, including the symmetric derivative of $a_2$.

The two metric occurrences in $B_8,B_9$ are inside $r_{\mathrm{D}}$. Their complete variation is $r_h$, giving ${\mathcal{M}}_8,{\mathcal{M}}_9$. The definition of ${\mathcal{J}}_{\mathrm{D}}$ fixes every coefficient entering $r_h$; no layer term is separately set to zero.

The last two branches have no metric argument. Their scalar slot derivatives give (eq:allscalar). The first nine branches have only one scalar occurrence and therefore have zero second scalar derivative.

For each mixed branch the two observable metric choices and two interaction scalar choices reproduce its four labelled matchings. For each scalar-sector branch the three observable scalar pairs and two bijections reproduce its six matchings. The count remains $9\cdot 4+2\cdot 6=48$.