Theorem: One fixed local free-product obstruction

There exist fixed real $f,\varphi ,\xi$ in the preceding classes such that

and

For $\lambda \ne 0$, this symbol does not descend through $q_0$. Its first-order coefficient also does not descend.

Proof. Choose a bump $\zeta$ supported in $B_{r_{\ast }}$, equal to one near $0$, and scalar data

Then ${\partial }_1\varphi (0,0)=1$. On a small relatively compact neighborhood of that event, ${\partial }_1\varphi >0$. Choose nonnegative nonzero $f$ supported there and a compact vector ${\xi }^{♯}$ equal to ${\partial }_1$ near $\mathrm{supp}f$. Thus $J>0$.

The metric representatives $0,K\xi$ are equivalent and both solve the vacuum metric equation. Their scalar labels agree. Lemma lem:poisson gives the displayed gap, which no function of their common quotient label can reproduce.