Theorem: Certificate preservation under a surjective dictionary

If $\delta$ is surjective, then

and

The complete left kernel is

Every such annihilator also annihilates $m^{\delta }$.

Proof. A surjective dictionary makes $A_{\delta }:{\mathbb{R}}^3\to {\mathbb{R}}^{|\mathcal{K}|}$ injective: each original coordinate is read by at least one code. Hence it preserves the ranks and the right kernel.

The original pivot columns become $8$ times the indicator vectors of the ${\beta }_{+}$ and ${\beta }_{-}$ fibers. Their annihilator equations are precisely (eq:pullback-left). They span the complete image, so the equations are sufficient. The target has coefficient $16/3$ on those two indicator vectors and is therefore annihilated by the same rows.