Lemma: Matching uniqueness in a fixed presentation

Suppose $A_a$ has full column rank. Then at most one complete output is accepted for that input. If an installed output $z_a$ satisfies all checks, it is the unique accepted output. Any candidate with a nonzero matching residual $A_{a}z-b_a$, or with a failed additional predicate, is rejected.

Proof. If $z,z^{\prime }$ both satisfy the affine equation, then $A_a(z-z^{\prime })=0$. Full column rank gives $z=z^{\prime }$. Additional predicates cannot create another solution of the affine equation. The rejection assertion is exactly the acceptance definition.