Theorem: Free-product obstruction

There exist fixed real

such that

and

For every real $\kappa \ne 0$, this fixed symbol does not descend through $q_0$. Its first-order coefficient does not descend either.

Proof. Choose compact scalar data at $t=0$:

where $\chi =1$ near the origin. The resulting free solution has ${\partial }_1\varphi (0,0)=1$, so this derivative is positive on some relatively compact spacetime neighborhood $U$.

Take nonnegative nonzero $f\in {C_c}^{\infty }(U)$. Choose a compact gauge vector equal to ${\partial }_1$ near $\mathrm{supp}f$. Then $J=\int f{\partial }_1\varphi >0$.

Both $0$ and $K\xi$ solve the vacuum metric equation, and they have the same quotient class. Their scalar labels agree. Since $w_{\mathrm{P}}(K\xi )=-\xi$, equation (eq:product-gap) follows. A descended function cannot distinguish these representatives.