The fixed-kernel decision problem

For one fixed input package, the object is

Here $U$ is a relatively compact joint-coordinate patch, $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ has integral one, and the internal kernel is kept exact.

The target is a statement on every compact joint test, including tests meeting the aligned-null locus $N$. No restriction to factorized witnesses or normal annuli is made.