Will this be a case of self plagiarism or will it annoy the referee?

Question

About 2 months ago, I uploaded a fairly long paper (P1) to arXiv and it is currently under review.

Now, I am writing a second paper (P2) on a somewhat different topic. But quite unexpectedly, it seems that to justify a single small step in one section of (P2), I will need a result that I proved in (P1). Unfortunately, this result in (P1) needs several lemmas before it can be proved and the proof of it takes up about 5 pages in (P1).

I have two choices:

1) Do the small step in (P2) by merely citing the statement of the result from my recently uploaded arXiv paper (P1) : But as I mentioned before, the step for which I need this result is a rather small one. Would it really annoy the referee (eventually when I submit this paper) if he/she is referred to 5 pages in a long paper on arXiv merely to justify a small step?

Ideally, I would have liked to keep the papers disjoint, but it does not seem like I can get rid of this one small step in paper (P2).

2) Copy down the entire proof of the required result from (P1), probably in slightly abridged format. But even then, this is likely to add at least 3 pages to the current paper and so it feels unethical (like self plagiarism / salami slicing or something like that).

So what should I do? Are my worries about option 1 (annoying the referee) unfounded? If you were a referee, would you be irritated if I referred you to several pages in a long paper to justify one small step?

Thanks

Answer 1

I have two choices:

1) Do the small step in (P2) by merely citing the statement of the result from my recently uploaded arXiv paper (P1).

Yes, do that.

But as I mentioned before, the step for which I need this result is a rather small one.

I'm not sure exactly what "small" means here. If you mean that it would be easy to get around this step by doing something else, then sure: definitely consider doing that. But given that you're contemplating attaching three pages of text, it seems that can't be right. If you mean that this part of your argument takes only a small amount of space in your latter paper: so what? If you mean that this step only contributes a small part to the result, then perhaps you might want to state the weaker result which does not use this step and then follow with a remark that using your other work the result can be improved.

Would it really annoy the referee (eventually when I submit this paper) if he/she is referred to 5 pages in a long paper on arXiv merely to justify a small step?

I don't see why that would be annoying at all. Virtually all significant contemporary mathematical work builds critically on past work. Referring to the arxiv is ideal in one sense because everyone can access it: it is so much better than referring to your unpublished PhD thesis for the proof of a "technical lemma" (which happens frequently, by the way). It would be easier for the referee if the other paper had already been published, but so what: things can move quickly, and most people do not wait for their past papers to be published in order to use those results. Because so many other people -- including leading people, whose work receives a lot of attention and is probably going to be essentially correct most of the time -- engage in this practice, by not doing so you would be slowing yourself down in a big way.

A conscientious referee would want to indicate whether or not she has read and vouches for the extra five pages or vouches for the result conditionally on the other paper. What to do depends a lot on the situation: if in those five pages you claim to prove the Riemann Hypothesis then she will really want to take a look. But reading five pages more is just business as usual for a referee.

2) Copy down the entire proof of the required result from (P1), probably in slightly abridged format. But even then, this is likely to add at least 3 pages to the current paper and so it feels unethical (like self plagiarism / salami slicing or something like that).

Although I don't recommend the practice, I don't think it's unethical in the slightest, so long as you refer to the other paper. Having your work be self-contained is a good thing. Having everything you want the referee to read placed under one roof is also a good thing. In this case I don't think that duplicating the material really saves anyone any time so I recommend against it, but you're not trying to gain any unfair advantage by doing it, so I have no ethical qualms about it.

Ideally, I would have liked to keep the papers disjoint,

Why?

Anyway, are you by any chance a student or junior researcher with a supervisor? This is a good question to talk to your supervisor about: there may be norms within your community or subfield that should inform your choice.