Proof Verification of Open Problems

Question

Over the past few days, I've noticed several proof verification questions regarding open problems. On one, quid left the comment:

I'm voting to close this question as off-topic because it asks for a review of a proof of a famous open problem.

There are some related posts on meta, but none of them directly address the question of what our policy on proof verification of open problems is, and I don't think the answer to that is self-evident.

The issue is primarily when the proofs are relatively long, like in this or this question. Such questions feel obtuse, in that an answerer will spend far more time understanding the question than in composing an answer - which means the question does not contribute to a Q&A, since anyone who stumbles upon the question would have to wade through it, and by the time they'd done that, they'd probably see the flaw.

On the other hand, questions like this present a very short proof, and I think there is value in having such questions, since they are easily understood, and the flaw in the proof is more widely applicable (i.e. not uncommon in other proofs we see here). A more borderline case is here - I chose to answer it because the mistakes are somewhat common and the length is manageable, although I can see an argument that it ought to be closed because the proof is poorly presented.

I don't think it would be wise to have a policy banning proof verifications asking about open problems, because they are, in spirit, within the realm of mathematics (i.e. can be effectively answered here) and not intrinsically different from the other questions of proof-verification - however, given that such questions often (but not always) include rambling proofs as a major part of their body, I think we need some clear policy on what to do with such problems, especially since the people asking such questions often ask multiple, and it would be thus wise to deal with their questions on a uniform basis.

What should we do with questions asking to examine a proof of unsolved problems?

Answer 1

The inspection of a long complex proof falls into the too broad category:

good answers would be too long for this format.

Which is a way of saying "you ask for more than this website can realistically provide".

As far as I'm concerned, the fact that the user seeks validation of a solution of a major open problem on a site like this one is enough evidence; no further consideration of the proof is needed.

Answer 2

The absence of a set policy regarding open questions seems reasonable, since questions about them (including naive attempts to prove) may entail nothing more than high school algebra or a simple misunderstanding of some aspect of the problem.

If a student accepts corrections, accepts that an idea is probably flawed, little is lost by helping. Sometimes it's just a matter of helping someone achieve clarity of notation, also I think a reasonable use of this site.

In many cases I think the user is not "seeking validation of a solution of a major open question" but simply asking why a particular elementary approach falls short. Not every two-page proof attempt is a quest for immortality and we should be able to treat different cases differently.

Answer 3

As a matter of principle I consider such Questions on topic. In practice of course the OP will not have given their "proof" sufficient critical thought to merit the effort of a critique.

Still, it's what I do. The famous problem need not be "open" for amateur researchers to have dreams of glory for their efforts, which often makes for a difficult job of convincing them their approach (and not only its detail) is hopelessly flawed.

I'm in favor of having our disposition of these Questions rest not on their "famous open problem" character but on the disposition of the Asker. If they are not willing to learn from their mistakes, close as "unclear what you are asking", or if the proposed proof is too voluminous as "too broad", or if the OP is unwilling to supply missing steps, as "lacking context or other details".