One divisor and the complete list

A divisor certificate checks

It does not certify that either $d$ or $N/d$ is prime. The root-list verifier in Lemma lem:verifier adds primality, ordering, complete product, and padding checks.

A bounded factoring recursion also needs its own worst-case control. Repeated successful splits have at most $\ell (N)$ prime leaves, but this does not bound the number of failed attempts unless those attempts are separately capped.