Pure-trace non-determination

This is a local proof of the probe non-determination shape recorded in CheungXVIIIA (locators :witness, :smoothwitness). It does not declare $B_A$ an authorized physical counterterm.

There is also a direct tensor sensitivity in the actual native rod term. Choose a spatial trace-free constant matrix $S$, for example $S_{12}=S_{21}=1$, and

with compact smooth $\rho ,\chi$. Then

Choose $\chi$ so that $S\nabla \chi \ne 0$; then $\text{calRz}\ne 0$.

For a second input $w$ with only

one has $\ell w=\eta g$. Its polarized local term with $z$ vanishes, but

A compact $g$ and output test $f$ can detect this expression wherever $\text{calRz}\ne 0$. Thus the complete native Hessian includes a mixed shear–lapse contribution that cannot be authenticated by a pure-trace calculation alone.