Proposition: Unconditional wave-operator family

Without using Input (S-bv), ${\sum }_{\rho ,\sigma }{\eta }^{\rho \sigma }{q^{\circ }}_{\{\alpha ,\beta +e_{\rho }+e_{\sigma }\}}=0$ on $\Omega$, since $\square S=0$ there (XVI-K, lem:grading(3)). The family spans a subspace of dimensions $(0,0,1,8,36)$, so $\dim {\mathcal{V}}_{\mathrm{adm}}\le 210$ without any kernel identification. With (S-bv) (certified), $140$ further independent relations appear.