Proposition: The rank report alone cannot decide an enlargement

Knowing only $\mathrm{rank}H=7$ and $\mathrm{rank}[H\mid b]=8$ does not determine whether an unspecified enlargement $E$ removes the obstruction.

Proof. In ${\mathbb{K}}^{15}$, let $H=(e_1,\dots ,e_7)$ and $b=e_8$. Both rank statements hold. Taking $E_{+}=e_8$ resolves the target; taking $E_{-}=e_9$ does not. For the second choice ${e_8}^{\mathsf{T}}$ annihilates both $H,E_{-}$ and pairs to one with $b$. These are synthetic examples, not loop images.