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We submitted a research paper to a journal for publication and we received the reports of the reviewers. However, the two reviewers pointed out a flaw in one of our proofs. Unfortunately, we cannot find another proof and we cannot also find an exemple in which the result doesn't work. One of the reviewers said that we can remove the result. The editor-in-chief asked us to send the revised version as new submission. What we can do in this case?

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Depends on the journal. Best is to ask your supervisor or a senior scientist who has experience publishing in this journal. – gerrit Mar 15 at 14:45
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If you cannot actually prove the result then you obviously cannot include it as a result. Whether you want to formulate it as a conjecture or not is up to you. – Tobias Kildetoft Mar 15 at 14:46
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I see four options: (i) Retract your submission entirely. (ii) Remove the faulty theorem and proof. (iii) Remove the faulty proof, but leave the theorem as a conjecture. (iv) Fix the proof (and possibly also change the theorem accordingly). Which option is best depends on your exact situation. Talk to someone senior who is familiar with the paper or subject area. – Thomas Mar 15 at 16:15
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It seems important here what your area of research is. If it is any type of pure mathematics, then a statement that has no known proof absolutely cannot be called a theorem, and there's no such thing as an "incorrect" proof because if it's incorrect then it's not a proof. If rather this is in the applied sciences, maybe you have some experimental evidence and you could talk about the statement in that way (but that's not my area so I don't know). – j0equ1nn Mar 15 at 21:33
    
@j0equ1nn, there is experimental evidence (sort of), OP tells us they couldn't find a counterexample. Maybe it should be published as a conjecture, with said evidence. – vonbrand Mar 16 at 1:26

First, notice that the question asked in the title has already been answered: your paper has not been rejected solely because an incorrect proof has been found. So your real question seems to be "What do I do now?"

The obvious answer seems to be to do what you have been advised: resubmit the paper with the faulty proof removed. To me an "unproved theorem" sounds a bit self-contradictory; although there are some things that you might want to informally describe that way, using that language in a paper seems to be asking for trouble. Rather you should decide whether you want to include the statement that you now realize you cannot prove in some form, e.g. as a question or a conjecture. If you think the work would be much more valuable if the statement was proved, you might want to delay resubmission while you make a more concerted effort to prove it. Or you may decide that without the theorem you had, the paper is not in your opinion worth publishing. All this is up to you.

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Thank you for your answer. As you know, our paper will go through another round of peer review, and might not be evaluated by the same reviewers since we have to send it as new submission. So I was asking if removing the theorem will lead to rejection. The reviewer ask to exclude the theorem as unproven and not to state it as unproven (I correct my initial question). – Farah Mind Mar 15 at 15:18
    
@Farah: Your edit, "One of the reviewers said that we can exclude the theorem as unproven." is still not completely clear, but I think you mean that you have been asked to remove the result. You should certainly not call something a theorem which you have not proven, but you could make it a conjecture or a question if you want.... – Pete L. Clark Mar 15 at 15:32
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In terms of whether your paper will be rejected: no one knows whether a paper will be accepted or rejected until it happens. In that you got asked to remove the theorem and resubmit, some experts have indicated that the paper still has some value without the theorem. The editor and the referees have so much more information than we do: they know your academic field and subfield, the journal submitted to and they've read the paper. Their guess is better than ours. If you are student, you should talk to your advisor about this. But you are not going to get a definitive answer. – Pete L. Clark Mar 15 at 15:34
    
You are right, we cannot call something a theorem if it has not been proven. The second reviewer said only that the proof is not clear. My advisor said that we have to prove the result, but I am a PhD student, I have not so much time and the first round took one year. – Farah Mind Mar 15 at 15:56
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@FarahMind, whether you have time or not to work on it does not matter to the journal nor to the reader! Either fix the flaw in the proof, or state it as a conjecture (with honestly confessing that you do not have a complete proof yet). Don't try to get your work published by any means. Yes, the math journals have long turn around time and your resubmitted paper may probably be rejected in the end. But if you manage to publish this work, and someone else in future shows that the proof of the theorem is flawed, it will harm your career far more than a rejected paper)! – John Mar 15 at 16:38

You are worrying about a superficial consequence of doing mathematics research instead of what really matters, which is the research itself. This kind of thinking is an example of what's known as "putting the cart before the horse", and is in general a recipe for trouble.

What you should do is to do your best to try to prove the conjecture you thought you had already proved. If you succeed, great. If you fail, go back and revise the paper to reflect the new state of affairs. Treat this writing assignment as a completely new and independent project to the one you had before you discovered the flaw in the proof. Right now you seem to be emotionally attached to the idea of mentioning the "theorem" in your paper, but you should let go of that notion: you should decide on whether to mention it or not by taking a fresh look at the situation and asking yourself whether as a conjecture it really deserves to be mentioned, in the sense that it truly serves the interests of the readers and of the mathematical theory you are working in.

The bottom line is, don't second-guess the reviewers' advice. Just do the best job you can do to write the best paper you can, resubmit, and hope for the best.

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Yes: in OP's practical world, this boils down to calmly addressing the academic publishing process and dealing with things in flux. Be conscientious, receptive, adaptable and patient. Also, be grateful for reviewers willing and able to work for nothing, finding holes in stuff that you might have ended up having to look like an idiot defending in print. Go round again... and again... and hope for similarly competent input on each circuit. Observably I do media/criticism stuff, but I do understand pure science, and I often wish that 'my' community could rely on this kind of scrutiny. Use it. – Captain Cranium Mar 16 at 0:24

What's the flaw in the proof? That can really matter. Have you looked at adding any extra assumptions which are required in order to make your proof work? If you can find such assumptions, you can include the proof of this weaker statement, and then have it as a conjecture that the result can be generalized (and give reasons/examples).

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In fact the mistak was in writing the theorem and not the proof. The proof we have sent treat a case, and the theorem is about another case, we added unintentionally one letter that changed the hole theorem. For the reviewers , the proof is not clear since it doesn't treat the case of the theorem. Now, we are trying to prove the case that we mentioned in the theorem. – Farah Mind Mar 15 at 22:47
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If you have other results, it may be better to just edit the theorem so that way it's just about the case you proved, and in the discussion note that further work should be to generalize the result to this other case (and now you're working on your next paper!) – Chris Rackauckas Mar 15 at 22:52
    
This is what we are thinking about, but the case that we proved is less importante than the case mentioned in the theorem. – Farah Mind Mar 15 at 22:56

If you have no proof, anything could be the case. You could formulate a conjecture, this is legitimate. However, you have no idea whether it is true or not, and, as such, this is not a theorem. If you can show independence, you can take it (or its opposite) as axiom. However, hoping to getting an unproven "theorem" published is not serving science, and, by extension, not helping yourself, either.

If you consider the paper without this conjecture as too weak, you could collect more results before resubmitting - however, the editor recommends resubmission, so it does not sound as if all is lost.

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