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I recently refereed a paper that I returned to the author(s) for revision. The thrust of their argument relied on a claim whose justification I felt was lacking. I dutifully raised the issue in my report and, in addition, I corrected another portion of their proof.

The author(s) have yet to revise their work and, in the interim, I came up with a justification for their claim. I now have a proof of this result (which is important for another paper I'm working on) and would like to publish it.

What are the ethics/options here? Do I need to provide them with the correct proof? May I submit the result as my own after their revision? Should I recuse myself from serving as the referee?

EDIT: Not sure if this makes a difference, but the conjecture the author(s) purports to resolve is one that I raised in a previous a paper and one for which I obtained partial results with similar methods. Although, I have worked on the problem and obtained partial results, I would not have obtained a proof without refereeing their work.

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    $\begingroup$ You might get in touch with the editors about this issue. Some unknowns: do the author(s) agree that the justification was missing, or just that you're requiring more details? is this claim very "visible" in the paper, e.g., stated in the introduction, or hidden inside? how much time do you need to expect a revised version (the editors can help you since this depends on whether they already transmitted the report, and when)? do the author(s) deserve credit for the bare statement of the claim (even with incomplete/flawed proof)? etc. $\endgroup$ – YCor Nov 1 '19 at 21:24
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    $\begingroup$ Although surely the paper itself was about research mathematics, it seems to me that this question is not, and that neither are mathematicians particularly qualified to answer it; so that it probably better belongs on ASE, which anyway many of our regulars frequent (and is quite active). $\endgroup$ – LSpice Nov 1 '19 at 22:09
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    $\begingroup$ But the "ownership" of a theorem, of a lemma, the distinction (who stated it)/(who proved it), is something somewhat specific to mathematics. $\endgroup$ – YCor Nov 1 '19 at 22:17
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    $\begingroup$ I actually like the questions, and I would not close it. Many of us have to deal with such problems. Knowing opinion of other people could be helpful. $\endgroup$ – Piotr Hajlasz Nov 1 '19 at 22:30
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    $\begingroup$ "May I submit the result as my own?" Wow, I would feel pretty bad for the original authors if their paper gets scooped by a referee. I usually try to trust my referees to not do that. $\endgroup$ – Zach Teitler Nov 2 '19 at 3:52
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The answer to your question really depends on how much work required justifying their claim. In my personal experience:

  • It was many times that a referee provided me with an argument that lead to a simplification of my poof or even provided me with new results that I included in my paper.
  • I did the same many times when I refereed the papers.

Therefore if the justification did not require much work, I would give it to the authors of the paper and let them publish it. Then you would be mentioned as an anonymous referee.

May I submit the result as my own? If so, must I wait until their revision before I submit my own paper?

That I do not really understand. Since they do not have justification of the claim, they cannot publish it. If they come up with a justification, your proof is of not much value since they already have a proof. In my opinion, you must not publish your proof before their paper is published.

If proving the claim is really difficult and required a lot of work, I do not really know what to advise. In one related situation I did as follows:

I was a referee of a paper $X$. However, before refereeing the paper I already knew how to prove a much better result (5 pages of elementary calculations vs 40 pages of a difficult proofs of far better results). But I only had my proof in a draft. I wrote to the author that I was a referee and I suggested that he would withdraw the paper from the journal and we would publish a joint paper. This is exactly what happened.

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    $\begingroup$ In my opinion, you must not publish your proof before their paper is published. - If the OP was already independently working on this, I see no good reason why the OP needs to wait. $\endgroup$ – Kimball Nov 2 '19 at 7:54
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    $\begingroup$ I agree that the situation when the amount of work needed to fill the gap is small is the most clear cut, in which case I would agree with the above answer. Small results are obtained independently all the time, so I see no problem with the referee putting the proof in one of their own papers. Then it also wouldn't hurt to refer to the unpublished work (the one under review) as an independent appearance of the claim (even if without proof). That assumes, of course, that the unpublished work with the claim is publicly available (e.g., on the arXiv). $\endgroup$ – Igor Khavkine Nov 2 '19 at 12:15
  • $\begingroup$ @IgorKhavkine I do not understand your answer. You say you agree with my answer that the referee should let the authors publish his proof. Then you say something about refereeing to the unpublished work under review. What work? If you did let the authors publish your proof there is no unpublished work. Am I missing something? $\endgroup$ – Piotr Hajlasz Nov 2 '19 at 20:05
  • $\begingroup$ @PiotrHajlasz Let me write this in symbols. :-) $U$ is the work under review, $R$ is the work that the referee would like to publish, while arrows are citations. I'm saying that it's OK to have any of the following scenarios: $U($claim$)~~R($proof$)$, $U($claim$)\leftarrow R($proof$)$, $U($proof$)\leftarrow R($proof$)$, $U($proof$)\leftrightarrow R($proof$)$. Realistically, there will even be enough time for the arXiv versions of both papers to go through exactly these stages before both are published. $\endgroup$ – Igor Khavkine Nov 2 '19 at 22:06
  • $\begingroup$ Just to clarify, I think that upon publication the final state should be one of $U($proof$)~~R($proof$)$, $U($proof$)\leftarrow R($proof$)$ or $U($proof$)\leftrightarrow R($proof$)$, with any of them acceptable, but probably the last one preferred. Of course, all of this is still contingent on filling the missing proof to be a small amount of work. When it is not, there's probably no universal answer. $\endgroup$ – Igor Khavkine Nov 2 '19 at 22:11

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