Corollary: Invariant quotient and tests

The algebraic class of $F(c)$ in $E_{\mathrm{out}}/{\mathcal{A}}_{\mathrm{phys}}$ is independent of the compatible prescription. A linear test $\ell$ is invariant under all compatible changes exactly when

Proof. Two solutions differ by an element of $\ker L$, and their coefficients differ by its image under $B$. The test is invariant exactly when it annihilates every such difference.