The next multiplication fails the ordinary test

The cross-line factor in the actual target remains grouped as ${H_T}^{\otimes 2}-H^{\otimes 2}$. The preceding audit does not expand it into mixed $H$–${G_{\mathrm{act}}}^{\mathrm{A}}$ products or replace it by $V$.

In particular, the audit is not a proof that the complete grouped coefficient has the displayed wavefront direction. A cancellation theorem for that coefficient would have to address the complete sum before claiming such a direction is present or absent.