Theorem: Complete evaluated contact image

and

The original minor on rows two and three and columns ${\ell }_a,{\ell }_b$ is $64$.

Proof. The product in (eq:BT) reconstructs every column in (eq:full-matrix). Its two left-factor columns are original columns of $N_S$. Their last two entries form

whose determinant is $64$. This gives the lower rank bound over both fields. The zero first row, or the factorization, gives the upper bound. The same two columns span the entire zero-first-entry plane.