Conjugate-sector coverage

Conjugation preserves functional support on the unrestricted real configuration space, since vanishing of a dependence is equivalent to vanishing of its complex conjugate. The same statement holds for its usual complexification.

Let $S(d)={\star }_c(d)$ and

The inherited reversal identity gives

Indeed,

Thus a complete commutator-vanishing proof on the original first-stage pairs transfers to their declared conjugates. This transfer is explicit; it does not supply the missing proof on the original pairs. Nor does generatorwise vanishing evaluate all actual linear relations among prescribed tensors.