With regard to the Marcie Rathke affair, a number of readers have seized on the provisional nature of the acceptance letter from Advances in Pure Mathematics. Indeed, they did not accept it outright, for the referee says that certain revisions are needed: rewrite the abstract, explain the notation, include proofs of the main result and key lemmas. Some said that since these revisions would either be impossible or would result in a totally different (non-nonsensical) paper, that this lets APM off the hook. Others suggested that this sort of “acceptance” was actually a rejection intended to let the author down more gently.
If so, it would be completely at odds with the review practices that are usual in mathematics.
Normally, provisional acceptance means that the referee thinks the paper’s topic is interesting and significant enough to be worth publishing, and that any errors or deficiencies it may contain can most likely be fixed. The referee may point out mistakes or gaps in the paper’s logic, or places where something is not clearly explained, and ask the author to correct them before the paper is finally accepted. In the vast majority of cases, the author is indeed able to correct these errors (perhaps at the cost of adding extra assumptions, or weakening their result slightly), and the paper goes on to be published. Occasionally it may turn out that an error is more serious and actually invalidates the whole paper, but this is rare; if the referee could tell that the error was so fundamental, the paper would have been rejected outright.
Of course, in this case, the referee couldn’t really have thought the paper was interesting or significant, because it was utter garbage from beginning to end. He or she made no claim to understand it in the slightest, and suggested generic improvements that couldn’t possibly result in a genuinely publishable paper (but might have inflated the page count into a higher fee bracket).
The idea that provisional acceptance might be used as a “gentle” rejection is also implausible. Experienced reviewers know that if you receive a seriously deficient submission and recommend provisional acceptance after major revision, the author won’t interpret this as rejection but rather encouragement. You’ll soon receive a revised manuscript in which a few small changes have been made, but the paper is still unpublishable. This could be iterated indefinitely, wasting an arbitrary amount of everybody’s time, until you finally give up and reject the paper. So if you don’t seriously believe that the paper’s flaws can readily be fixed, you reject it at once.
Regarding the Rathke paper, my belief is that a few tweaks, the addition of a few more randomly generated definitions and proofs, and of course payment of the processing fee, would have led to the paper being accepted and published. As I’ve explained, for various reasons, I wasn’t really motivated to pursue it beyond the initial (provisional) acceptance, and so I suppose we’ll never really know what would have happened. But I think, and I think most of the mathematical community would agree, that the point has been made.