Why one nonzero response is insufficient

At ${\beta }_{+}$, the target is $16/3$ while the allowed raw response $n_a$ is $8$. The scalar contact image at that row is already all of $\mathbb{R}$. The second nonzero row is independently spanned by $n_b$. The blind row imposes no condition on either the target or the contact image.

This explains the finite rank: all columns lie in a two-dimensional row plane, and two original contacts span that plane. The proof does not require printing $45360$ integers. Complete reconstruction and an original minor supply exact coverage and both rank bounds analytically.