First of all, if we would like to have a specific rule against such questions, I don't think it is useful to make the topic "P vs NP" part of that rule. I don't believe we are being flooded by questions about that topic and I don't think that that topic is inherently
'problematic', nor that it is within the domain of research. (That is, questions about the open question (and how to deal with the fact that we don't have an answer) are central to the field of complexity, which permeates to all sorts of 'CS experts')
I think I agree mostly with this answer, although that has bit of a different focus. However, the following two reasons to distinguish good from bad questions on this topic mentioned there are relevant:
they are honestly phrased as people trying to understand what they are missing,
they are rather short, have a few points, and are easy to answer for an expert in complexity theory.
I think point 2 is key here. There's quite a difference between someone asking to check their 40 page proof versus someone wondering why an elementary proof of 4 lines doesn't solve this open problem.
Point 1 seems more a case of etiquette than a meaningful property of the question. So what if someone really their crisp and clear 4 line proof of P=NP is correct? As long as the question is clear and they behave themselves (in particular when proven wrong), I do not see why this makes the question bad.
A more relevant criterion, that also focuses more on the question, rather than on the user is
- the question contains a cleanly phrased, well-formed proof based on standard terminology
I think this covers all cases of questions we do not want that are excluded by point 1. Point 1 seems to be aimed at 'cranks' that are prone to long, vague proofs littered with (often undefined) non-standard terminology. The long part is already covered by point 2, the vague and non-standard terminology is covered here.
... this is not peer review, [...] this is [the] not proper place for scientific advancements and in my opinion there is no chance for breakthrough here.
Is this not the proper place for scientific advancements? Well..., why not? But more to the point, I do not think 'scientific advancements' is the value of the good questions about these topics. The value is that these wrong proofs often make assumptions that are unfounded, but seem very intuitive. (the mother of all assumptions is of course, "clearly, we cannot do better than brute force" such as in this question) It just so happens to be the case that the central questions in complexity have remained open problems for a long while, despite appearing relatively simple. It is good for e.g. a student to be aware of that.
Put more poignantly, the error in these proofs is often educational, even for others than the one that has written the proof.
I am puzzled, because questions "where is error in my attempt" seem to be good, but very similar questions with different wording get downvoted and closed. Posts with hidden attempt to prove get closed.
I understand your confusion, allowing questions on (complete) P vs. NP proofs seems counter to the policy of not checking peoples answer. I think that point 3 here denotes a clear enough boundary here. If the proof itself is incomprehensible, poorly structured, or has other significant problems unrelated to the subject matter, then this person is better served by improving their basic proving skills before looking into proofs of this complicated matter. These are off-topic, and should be closed.
Note that also, even if the question literally asks to check their proof, it should be interpreted as 'find the mistake'. This is different from grading an assignment, even if we are guaranteed to have a mistake somewhere. Why? An assignment is a measure of the writers competence in their field. An error in an assignment is basically everything that shows incompetence (In practice, grading schemes are of course more lenient and require the student to only show competence on a few well-defined points). We are not here to judge/grade someone's competence. But I see no reason to disallow questions about erroneous arguments for that reason.
To clarify my viewpoint, let me give a few examples.
Naive argument that P ≠ NP: this question gets a full score, passing all three points. I think this is a good question and clearly so.
Contradiction proof for inequality of P and NP: this one passes 2 and 3, but it is not clear to me whether it passes 1. This is one of the reasons that I prefer to judge only on points 2 and 3. I think this question is fine and well within the lines, although not as clear as the previous one. (The difference in votes is better explained by the HNQ exposure of this question, rather than the inherent quality)
Solving Diophantine equations: does having a bound on the size of the solution help?: Note that this question has been rather radically edited to solve various issues. In its original form, this is a question that I would consider a bad one. It is long, and littered with irrelevant or mostly unhelpful digressions. On the surface, it fails all points, and in its original form, it should have been closed. Still, it is based on an interesting assumption, which is not clearly false, but unproven (and likely even harder than resolving P vs. NP). This is not obvious at first, however. Unless someone is able to 'extract' a good question out of these, these questions should be closed. While it is nice if someone does this, we have no obligation to the author to do so.
In summary, I think that asking about such proofs should be acceptable, if
- The proof is short and has only a few points; and
- The proof is cleanly phrased, well-formed and based on standard terminology.
- The question has no other problems that would apply to any type of question. See e.g. here or the help center for more on what to do and what to avoid when asking a question.