Bounded error and exact output

Shor supplies a foundational polynomial-time quantum factoring algorithm with a randomized success guarantee Shor. That is a cited precursor, not a proof of the frozen fixed-alphabet probability-one claim.

If an independent trial succeeds with probability $s<1$, $M$ independent repetitions still all fail with probability

at every finite $M$. A retry-until-success strategy may halt almost surely and have finite expected time, while having no finite worst-case bound. These are different claims.

Exact algorithms in other models also require their gate and input assumptions to be matched; computed-angle exact Fourier transforms do not automatically give an exact circuit over the frozen alphabet MoscaZalka.