Proposition: Approximate and finite-probe cancellation need not be exact

There is a continuous linear repair map with dense image for which every finite coordinate test can be cancelled exactly, while the full defect cannot be cancelled.

Proof. Let ${\mathcal{V}}_{\mathrm{def}}={\ell }^2(\mathbb{N})$, let ${\mathcal{C}}_{\mathrm{ct}}=c_{00}$ carry the inherited ${\ell }^2$ norm, and let $B$ be the inclusion. Its image is dense but not all of ${\ell }^2$. Take

Then $D\in {\ell }^2\setminus c_{00}$, so no $c\in {\mathcal{C}}_{\mathrm{ct}}$ satisfies $D+Bc=0$.

For the probe onto the first $m$ coordinates, choose

It cancels those probes exactly. Moreover,

Neither fact puts $-D$ in the actual image.