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This post is made in response to this thread, which asked for the error in a purported proof of the Riemann Hypothesis. After an upvote/downvote war (the downvotes have currently narrowly won), it was closed after receiving an ultimately popular answer (mine) for being off-topic.

The moderator @XanderHenderson made several specific claims about what is and is not on-topic. I find those claims inconsistent with the on-topic and off-topic FAQ's. For example:

  • The FAQ: "There are certain subjects that, while still on-topic here, might be better addressed by one of our sister sites: Research-level mathematics ..."
  • Xander: "Research level questions are not, generally speaking, on-topic here."

and

  • The FAQ: "Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much."
  • Xander: "Math SE is intended to be a repository of narrow, focused questions with high-quality, authoritative answers"

and

  • The FAQ: (nothing)

  • Xander: "we cannot peer-review the work of others"

This post is not specifically about Xander's claims or this one particular case. (To be clear, I respect Xander and entirely believe he's acting in good faith.) He concluded with "please note that I was elected to interpret policy on this site, and to enforce that policy".

My issue has to do with this "interpretation" step. The two linked FAQs are quite vague and leave a lot of room for individual variation on exactly how they're interpreted. Xander has a rather narrow interpretation (which again I find inconsistent with what's written), whereas I have a rather expansive one. It's entirely unsurprising that different experienced users will disagree given the current state of the FAQ's. That inevitably leads to somewhat arbitrary votes to close for vague or inconsistent reasons.

So, my questions: why are the FAQ's left so vague? If Xander or another moderator's actual interpretation is law, why aren't the FAQ's simply updated to reflect that law? More broadly, what can be done to improve the consistency and usefulness of what is deemed off-topic?

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    $\begingroup$ I think I'm not the only one here that sees no contradiction at all between either of the first two bulleted pairs. There may be something interesting to say about the third pair. $\endgroup$
    – rschwieb
    Jul 14 at 18:41
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    $\begingroup$ Claimed proofs of the Riemann Hypothesis are generally unwelcome at mathoverflow. One recent case ended up at skeptics.se because it could not find a home at either mathematics site. skeptics.stackexchange.com/questions/51965/… $\endgroup$
    – Henry
    Jul 25 at 19:02
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TL;DR

Addressing the question at the end of your post, the FAQ is intentionally left vague because (a) the guidelines are, to an extent, network-wide boilerplate without a lot of room for per-site customization, (b) the FAQ is not easily changed, and (c) community standards are meant to evolve over time and the FAQ is not meant to interfere with shifts in scope and emphasis. More detailed guidelines, policies, and procedures are ironed out via meta discussion. This is by design.

Now, addressing your purported contradictions point-by-point:

Peer-Review

  • The FAQ: (nothing)

  • Xander: "we cannot peer-review the work of others"

The job of a peer-reviewer is to scrutinize a paper and check the details. It typically takes me many hours, spread out over a week or two, to peer-review a paper for a journal (others may work faster or slower---I can only describe what is typical for my own workflow). Peer-reviewing a paper is (or should be) an intense activity. Math SE is not setup to provide this kind of scrutiny, even if a paper is so deeply flawed that a potential answerer can spot fatal errors in the opening sentence.

As a general policy, requests for peer-review are far too broad for Math SE, and are, therefore, off-topic.

Scope

  • The FAQ: "Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much."

  • Xander: "Math SE is intended to be a repository of narrow, focused questions with high-quality, authoritative answers"

Again, I see no contradiction here. Questions should be focused. The FAQ gives an extreme example---a question which requires a book to answer to way too broad. However, a question can be too broad but not require a book to answer.

Generally speaking, questions which present an argument then demand that MathSE users "spot the error" are too broad. This is true even of the tag (which, by the way, is not a tag without controversy): a good proof-verification question should discuss specific questions about the correctness of a proof, and should focus on specific uncertainties that the author may have.

From the tag-wiki:

This tag should be used when you have a proposed solution to a problem that you want to have looked over by somebody else. It is useful for these questions to include an explanation for why the argument they present is not convincing enough. Further discussion on using this tag can be found in the Mathematics Meta post here. Questions of the type "I have written this proof, is it right?" are, typically, too broad.

Also relevant is the meta thread The problem with proof verification and the links in the question. Keep in mind that policy on Math SE is meant to be arrived at collectively---perusing those threads, I come to the conclusion that the tag is meant for questions about proofs written by the question author, where the author is seeking clarification regarding specific issues in their argument.

All of this is doubly true when the proof being verified is a claimed proof of a major open problem.

Research-Level Questions

  • The FAQ: "There are certain subjects that, while still on-topic here, might be better addressed by one of our sister sites: Research-level mathematics ..."

  • Xander: "Research level questions are not, generally speaking, on-topic here."

I see no contradiction here. The FAQ states that certain topics---specifically, research level questions---are more appropriately asked on MathOverflow. The general SE model asserts that any particular question should only be asked on one site in the SE network (as a general rule cross-posting is discouraged). Ergo if a question is more appropriate on one site than another, it is out-of-scope or off-topic on the site where it is less appropriate. Therefore, as a general rule (and not as a hard-and-fast stricture), research level questions are not on-topic here, as they are on-topic on MO.

The line between the two sites is fuzzy, but, speaking in generalities, research-level questions don't belong here, and will often be migrated to MO. The goal is to find the right home for any particular question, and I don't imagine that a question would be closed on Math SE simply because it is research-level---again, migration is a more typical outcome. Of course, if a research-level question is otherwise fine, and is not welcome on MO, then it is likely that it will have a home on Math SE.

Regarding the Specific Question Discussed Above

The question https://math.stackexchange.com/q/4197244/ is off-topic on MathSE for a number of reasons. The most salient problem with the question is that it is asking the Math SE community to peer-review (to "find the bug in") a paper written by a third party. Math SE simply isn't setup to provide this service.

Taking a step back, the question is also poorly-scoped. If the question were "I have read this paper, and am confused by (47) on page 23. How does the author arrive at this identity?", then we could reasonably expect the Math SE community to weigh in. In this case, any potential answerer is expected to read the entire paper---the focus of the question is not on any particular step or argument, so the question is too broad. Thus, even if the question were not a request for peer-review, it would be off-topic for Math SE.

Finally, and this is really the smallest flaw in the question, research-level questions are typically not on-topic here. The line between Math SE and MO can be blurry, and it is not clear that a reasonably-scoped question about this particular paper would be off-topic on Math SE, but, as a general rule, questions on active research topics belong on MO, not Math SE.

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    $\begingroup$ The question about the purported proof of RH would be closed quick smart if posted to MathOverflow, where there is a longstanding & well-respected policy of not responding to requests for reviews of papers and claims. MathStackexchange has generally been more relaxed in its reaction to that sort of question, but in any event suggesting that that particular question is more suited to Overflow will not please anyone over there. $\endgroup$ Jul 13 at 23:12
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    $\begingroup$ @GerryMyerson Which is why the fact that requests for peer-review are off-topic is the most important and prominent reason given for the closure of the question here. I did not suggest that this particular question would be welcome on MO, only that as a general matter, research level questions belong on MO, not MathSE. $\endgroup$
    – Xander Henderson Mod
    Jul 13 at 23:13
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    $\begingroup$ Thank you for taking the time to write such a detailed response. Unfortunately I still find it entirely inadequate. You somehow see "no contradiction" in verbiage that says "research level math is still on-topic here" and "research level math is generally not on-topic here". You then continue to expound on your interpretations of vague guidelines as if those interpretations were written in stone without even acknowledging that they're in no way canonical. All I'm asking for is clear policy, and all I'm getting is experienced users behaving in baffling ways. I think I'll just leave. $\endgroup$ Jul 14 at 8:24
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    $\begingroup$ @JoshuaP.Swanson Where do you see that anything I have said is written in stone? I think that I used a lot of hedge words, such as "as a general rule" and "typically". I also think that you are still focusing on the wrong thing with respect to the particular question here, which is a poor question first and foremost because it is asking for peer-review. The fact that it is a research level question is not doing the question any favors, but that is not the biggest problem with the question. $\endgroup$
    – Xander Henderson Mod
    Jul 14 at 13:22
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    $\begingroup$ Regarding whether or not my answer is canonical: policy on Math SE is determined through meta discussion. If my answer does not reflect the will of the community, then it will be downvoted, and it will not represent policy. On the other hand, if it is upvoted, it becomes policy. As any answer on meta has the potential to be policy, I prefer to write meta answers which are written as though they are policy. As it stands, my answer is currently at +4, which indicates that the majority of users who have bothered to vote thus far feel that my answer aligns with good policy. $\endgroup$
    – Xander Henderson Mod
    Jul 14 at 13:24
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    $\begingroup$ @Xander. both the question and answer here are so convoluted that I don't think it is wise to interpret anything from it. For example, I up-voted this meta post and down-voted your answer because solely your statement about research level question. This does not main that I agree the post on main is on-topic (obviously I think otherwise). $\endgroup$ Jul 14 at 15:56
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    $\begingroup$ You are a mod, you have lots of power in pushing policy. I saw that you've been mentioning that idea several times recently and I am very worried that one day you declare that research level question is off-topic here, however mild your current statement is. @XanderHenderson. $\endgroup$ Jul 14 at 16:28
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    $\begingroup$ @ArcticChar I am very upfront about my opinions about what is good or bad for the site. However, I try very hard to moderate based on consensus, rather than my own opinions. For example, I don't typically unilaterally close questions which include a genuine attempt (though I will happily cast a fifth vote on such questions). Regarding research-level questions: my usual approach is to discuss migration to MO. If it isn't on-topic there, but is otherwise a good question, I see no reason to intervene. $\endgroup$
    – Xander Henderson Mod
    Jul 14 at 16:52
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    $\begingroup$ It hasn't been my experience that research questions are off-topic here. I've asked multiple questions here which turned out to be research level and which were not migrated to MO; I'm not sure I've ever seen a question migrated from MSE to MO for being research-level without the OP explicitly asking for it, FWIW (maybe you have an example?). $\endgroup$ Jul 14 at 19:05
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    $\begingroup$ I think the problem here is a concerted effort by some users commenting below this post, to address matters not addressed in the original questions; essentially. The question linked by the OP is not anything close to a research question, so please don't hijack the question of concern: the OP's post. Their example is of a crappy post. $\endgroup$
    – amWhy
    Jul 15 at 18:45
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    $\begingroup$ @amWhy, it was Xander who introduced the question of MO vs m.se, and when a moderator writes about policy, that becomes a question of concern, regardless of what the original question of concern was. $\endgroup$ Jul 15 at 23:16
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    $\begingroup$ Correction: per site charter, the scope of this site is all levels of math, including research level. While some research questions might get better answers on MO, one should not apply any SE guidelines about cross posting since MO is not really an SE site (it just uses their software platform - see the contract between SE and MO). MO is completely independent of SE in other regards (except agreeing to protect confidentiality). $\endgroup$ Jul 18 at 23:28
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    $\begingroup$ I agree with the too-broad characterization of paper review requests. We should also apply this to proof-verification question that lack focus, i.e. don't explicitly say what part of the proof is in doubt, and why. $\endgroup$ Jul 18 at 23:33
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    $\begingroup$ @XanderHenderson, I appreciate your sharing the links, but your understanding is incorrect. Some questions can be on-topic on multiple sites. (For instance, there are questions that would be on-topic on both Computer Science.SE and Mathematics.SE.) That is not a contradiction to the rules on cross-posting. The rules on cross-posting do not imply that a question cannot be on-topic on two sites. Nowhere does that answer make such a claim, and it is not logically entailed. The rules on cross-posting state that even if a question is multiple sites, it is prohibited to cross-post it. $\endgroup$
    – D.W.
    Jul 21 at 0:07
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    $\begingroup$ @XanderHenderson, I wonder if I might politely suggest the possibility that you might be hearing similar arguments because you have made an inaccurate statement in the answer and have not edited the answer to correct it, and that this concerns various people. (I would dispute that you are hearing the same arguments; I believe no one has previously made the specific arguments I raised.) Rather than leaving a clarification in the comment, could I possibly encourage you to edit your answer to correct the incorrect statement? I suspect that to some people this point is concerning and not minor. $\endgroup$
    – D.W.
    Jul 21 at 0:21

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