Parity violation makes singular-value action basis dependent

For $A=0$, the entire domain and codomain are zero-singular spaces; both $u=1$ and $u=-1$ are unit left-kernel vectors, and $v=1$ is a unit right-kernel vector. With the first pairing, $\mathrm{\mid }1⟩⟨1\mathrm{\mid }$ sends $x$ to $x$, so the false zero-pair sum with coefficient $f(0)=1$ is $I$.

With the second equally valid left singular basis, $\mathrm{\mid }-1⟩⟨1\mathrm{\mid }$ sends $x$ to $-x$, so the same false rule gives $-I$. Since $I\mathrm{\ne }-I$ on $\mathbb{C}$, this rule is not basis independent. The valid odd definition avoids the defect because odd polynomials have $f(0)=0$ and the sum contains only positive singular values; the valid even definition adds $f(0){\mathrm{\Pi }}_{\ker A}=I$, which is canonical and unchanged by the choice of kernel basis.