Theorem: Permanent affine false-positive witness

Let

The old fallback $c_{\mathrm{old}}=(1,1,0{)}^{\mathsf{T}}$ has residual

The complete correct family is

Proof. Direct multiplication gives the residual and the corrected solution. The original minor on rows $1,3$, columns $1,2$, is $-1$. The complete left kernel is generated by $(-2,1,0)$. The displayed nonzero right-kernel vector spans the one-dimensional kernel. Apply Theorem thm:rank and Proposition prop:right.