Proposition: Arbitrarily small noise can destroy the certificate

For every $0<\epsilon <1$, there is a CPTP implementation perturbation of the finite circuit at half-diamond distance at most $\epsilon$ whose output on $N_0=12$ is incorrect with probability exactly $\epsilon$.

Proof. After the ideal circuit, apply

where $X_5$ flips output wire $5$. The two branches are CPTP. The half-diamond distance from the identity is at most $\epsilon$, by convexity and the maximum channel distance one Watrous.

In the ideal output, wire $5$ is the only one bit of the first prime entry $2$. Flipping it changes that entry to zero while leaving the later entries $2,3$ positive. The resulting word is not a valid zero-padded prime list for $12$. Thus exactly the perturbed branch is wrong, with probability $\epsilon$.