Proposition: Proposition 9, special-family 4D nonvanishing, m=0

On $U$, for the massless special family of Lemma lem:support,

Hence, for every $\xi \in {C^{\infty }}_c(U;T{\mathbb{R}}^4)$,

The integral is strictly positive whenever $p^{\prime }>0$ on $\mathrm{supp}\xi$ and $0\le {\xi }^2\not\equiv 0$. In particular $C{|}_U\ne 0$. The moment $m_0$ is positive because $S\ge 0$ and $S\not\equiv 0$.