# Submitting paper proving "X" soon after paper proving "X-epsilon"

I am a math postdoc, currently working on some conjecture. To avoid going into details, let's formulate it as follows: "every runcible doohickey is cromulent". A senior colleague told me last year that if I proved it, it would make for a "really nice paper" (that's her words).

Eight months ago, I found a partial proof: something like

Theorem 1: Every runcible doohickey is cromulent, unless it is in one of the families A_n, D_{2n+1} or E_6 (informally, this accounts for "less than half" of all possible families) and is larger than some minimum size.

(if you want details, the paper is here: https://arxiv.org/abs/1605.03833). I wrote it up and submitted it, to Journal A (a good but not top journal). It is currently still under review.

A few days ago, I finally completed the proof of

Theorem 2: Every runcible doohickey is cromulent.

(the paper is here: https://arxiv.org/abs/1612.08942). Following the suggestion of my colleague, I would like to try my chance and send it to Journal B (a top journal).

The problem is that the proof of Theorem 2 is really similar to the proof of Theorem 1; a lot of passages in the former are almost verbatim copies from the latter. (Of course I always disclose when it happens). There are still some significant differences:

• the definition of the central object of the paper had to be changed, adding a significant layer of complexity;
• a lot of intermediate proofs and definitions no longer worked, and had to be either considerably expanded or completely rewritten;
• paper 2 is generally cleaner and better-organized than paper 1 (despite the additional complexity);
• in paper 2, I proved an intermediate result that was not present in paper 1, and that maybe has some value on its own (though a colleague told me that in her opinion, it was just a trivial consequence of some results proved by our PhD advisor).

The issue becomes even more serious if you look at it backwards. Should paper 2 somehow get published before paper 1 (which is not completely ruled out given the vagaries of peer review in math), it would make paper 1 totally worthless, as almost every statement from paper 1 can be obtained as a particular case of a corresponding statement from paper 2.

(My feeling is that in an ideal world, paper 1 should have never been published. However I was in a rush to get published in time for job interviews, and I had no idea how long it would take me to get to Theorem 2.)

A few questions:

• Do you think there is any chance that, if the reviewer of paper 1 learns about Theorem 2, they will reject it, saying "this paper is not interesting, as it is just a particular case of Theorem 2"?
• Do you think it is likely that the presence of paper 1 will prevent paper 2 from being accepted by Journal B (which has high standards)?
• Should I go as far as to retract paper 1 before submitting paper 2? (However the reviewers of paper 1 probably wouldn't be too happy about this!)
• I am also considering the following course of action: submit paper 2 to journal B, and suggest that they talk to the editors of journal A and that they assign to paper 2 the same reviewers as for paper 1 (if they are so inclined). Is it acceptable to do such a thing? I see two reasons for doing so:

• The altruistic reason is to avoid wasting the reviewers' time. Both papers are quite long (73 pages for paper A and 89 pages for paper B), and someone who has already read A would have a MUCH easier time understanding B.
• The selfish reason is that it could speed up the review process (and help the paper get published in time for job deadlines).

Any other advice about what to do in this situation is welcome! My main preoccupation is to maximize my chances for getting a job; the relevant deadlines for publication acceptance are roughly March 2017 and March 2018. So far I have only two publications, both in medium to good (but not top) journals. I would especially appreciate answers from mathematicians, as I have the impression that standard practices differ significantly from field to field.

EDIT: my judgments about journal rankings were initially maybe too harsh (see comments below Pete C. Clark's answer). I did this mostly out of modesty, as I looked at it from my perspective and from the perspective of my potential employers; however I forgot to consider the perspective of the editors or reviewers of the journals themselves. I apologize for this little blunder; I corrected it.

• Based on the difference between Theorems 1 and 2, wouldn't it be a bit more accurate to say that you proved X/2 and then shortly afterwards X, rather than X-epsilon and then X? Commented Jan 2, 2017 at 0:15
• Well, you are right: calling it "epsilon" is maybe not quite appropriate. (Not sure if it is possible to change the title?) In fact, here is the way I see it. My "partial cromulence theorem" is actually the next-to-last step in a chain of maybe 7 or 8 generalization steps, starting from the construction of the first example of a cromulent doohickey (back in 1983) and going all the way to the "full cromulence theorem". The structure of the proof remains the same, but each step adds a little complexity. But it must be said that the last step is probably among the hardest. Commented Jan 2, 2017 at 1:18
• @IliaSmilga you can change the title, but based on your explanation I now think it is appropriate, or more precisely, strikes the best balance between being accurate and helping to make your question as interesting and general as possible. Commented Jan 2, 2017 at 3:57
• "Retract" is a strong word. You probably mean (and would be better off using) "withdraw", which is a lot easier on the ears.
– E.P.
Commented Jan 2, 2017 at 9:59
• If I were hiring math faculty, future interviews would certainly need to include assessment of familiarity with the partial cromulence theorem. Commented Jan 3, 2017 at 19:45

This is a great question, and one that I (and probably most mathematicians) feel sympathetic to. In most projects I've worked on, there is some tension concerning the issue of when to cut it off and call it a paper. Sometimes there is a lot of tension. (At least your papers are singly authored. When you are dealing with collaborators, most likely everyone feels that tension, but the forces imparted on the various coauthors are rarely identical and are sometimes all but antithetical.)

The first thing that I want to say is: take a breath and realize that it shouldn't matter too much in the cosmic scheme of things. This kind of thing is exactly why recommendation letters play such a large role in hiring decisions. You definitely want to get letters from people who will attest that you have proved the "Full Cromulence Theorem" rather than just the "Partial Cromulence Theorem". The more eminent your recommenders, the more trust you will be extended and the less pressure you have to "show your hand" at any point in time. But since you already have complete preprints available, in a purely mathematical sense you're done: you've proved the Full Cromulence Theorem and people will take that into account.

The second thing I want to say is: I strongly recommend that you talk to mentors and/or senior people in your field, including your thesis advisor and postdoctoral supervisor.

Okay, but since you asked: I don't think there's one clearly best way to proceed, and I think that you are choosing between things that everyone would regard as reasonable. You could absolutely withdraw the first paper from Journal A and submit the second paper to Journal B, explaining to both journals why you've done so. Then it's a good shot that they will get in touch with the referees of the first paper. Or you could do the same thing but not withdrawing from Journal A: it is not at all your fault that some months later you proved a better result. Finally, you could ask to replace your older submission to Journal A with your newer submission. But it sounds like by doing so you feel that you would be selling yourself short, which I find very reasonable.

Do you think there is any chance that, if the reviewer of paper 1 learns about Theorem 2, they will reject it, saying "this paper is not interesting, as it is just a particular case of Theorem 2"?

Anything can happen, but that would be a distressingly poor thing for the reviewer to say, given that s/he has already had the first paper for many months. I wouldn't worry about it.

Do you think it is likely that the presence of paper 1 will prevent paper 2 from being accepted by Journal B (which has high standards)?

This is a bit more likely, but I think the fact that paper 1 has not yet appeared works for you here. If paper 1 appears in Journal A, then in the minds of some referees and editors it could "put a lower selling price" on the Full Cromulence Theorem: someone can say that the real breakthrough occurred in the proof of the Partial Cromulence Theorem, and even that was only worthy of publication in Journal A, so the new mathematics added in attaining Full Cromulence is not worthy of Journal B. That sounds bad, and it is bad, but look: excellent journals can reject papers by making rather harsh decisions about their value. They do so all the time, and moreover they have to do so. It would be more worth your time to convince the community of the value of the Full Cromulence Theorem than by worrying about what the journals might do.

In fact it could also go the other way: if your community does not know what to make of cromulence theorems for doohickeys, then Journal B, upon receiving an 89 page paper on the topic, could say "The author is working way too hard -- and taking up way too much space -- to prove something of uncertain value. We're not sure that anyone cares about results of this kind." Well, if you publish a partial cromulence theorem in a mid-tier journal, then the community is clamoring for cromulence theorems. Just imagine how much more valuable a full cromulence theorem could be.

So you can't know. Prove the best theorems you can, promote them as best you can, and try not to lose sleep over whether your promotional campaign was the right one.

Should I go as far as to retract paper 1 before submitting paper 2? (However the reviewers of paper 1 probably wouldn't be too happy about this!)

As I said, you certainly could. If you do, you should explain why, and give the editors a chance to enlist the same reviewers for paper 2. But fundamentally it's your work, and you can do what you want with it, including withdrawing it. (By the way, I think it is not clear that the reviewers wouldn't be too happy about it. It is unfortunately possible that they spent very little of the eight months reading over your 73 page paper, in which case they could be quite relieved to have it off their plate.) I would say though that rather than definitively withdrawing the paper, it seems better to explain the situation to the editor and suggest withdrawal. At that point the editor may want to contact the reviewers, and if they are mostly finished the job and like the paper, then that may end up expediting the publication process.

Good luck, and congratulations again on proving that every runcible doohickey is cromulent. Full cromulence! I hope you are proud.

• I believe the technical term is "perfect cromulence". Commented Jan 2, 2017 at 13:23
• @Ilia: I'm really not sure, honestly. One comment: you don't know anything about the referees for paper 1 except that they have spent nearly 8 months with your paper already and haven't finished it. Since paper 1 is 73 pages, that's not too long by any means, but neither is it any evidence that they're acting with alacrity or are better than other referees in any way. Commented Jan 2, 2017 at 15:54
• As a side remark: I looked on your webpage, and "So far I have only two publications, both in low to medium-tier journals." I am not intimately familiar with Geometriae Dedicata, but I am pretty sure it is a quite respectable journal. For sure Annales de l'Institut Fourier is a very good journal. A low-tier journal means something else entirely. Commented Jan 2, 2017 at 16:07
• @IliaSmilga I think the idea in your fourth bullet point is very good. It is not a common thing to do, but that is beside the point. Your intentions are honorable and your logic for why this makes sense is very solid. In your email to the editor of journal B, be polite and make sure to phrase this as a suggestion and make clear that you accept the editor's decision to act as they see fit. The worst thing that can happen is they'll ignore the suggestion. Commented Jan 2, 2017 at 17:22
• @Ilia: I am impressed that you have a publication in Annales de l'Institut Fourier. Anyway, pro tip from one pure mathematician to another: f\$\cup\$ck impact factors. They are not what we use to evaluate journals. Commented Jan 2, 2017 at 19:32

Any other advice about what to do in this situation is welcome!

Well, you are in pretty good company -- the best, in fact -- for proving X soon after proving X minus epsilon. Albert Einstein published the main ideas for the special theory of relativity in his famous paper On the electrodynamics of moving bodies dated June 30, 1905... except that the theory was incomplete, since he had not realized its implications for mass-energy equivalence - the famous equation E=mc2, which nowadays everyone agrees is a crucial part of the theory. So he published that part soon afterwards in a second paper, Does the inertia of a body depend on its energy content?, dated September 27, 1905.

Oh well, progress in science is just messy I guess. Einstein managed to get a job based on his Annus Mirabilis papers, though I seem to recall that it took him a few years.

As for what to do, Pete Clark's answer is just cromulent.

• I will add that I'm familiar with other examples --- much more recent and in mathematics --- of authors proving the full version of a result mere months after releasing partial versions, and publishing both. Commented Jan 2, 2017 at 16:57

Is the Partial Cromulence Theorem considerably easier to use and/or understand than the Full Cromulence Theorem? Does it cover a large fraction of the potentially useful applications? (Your "X-plus-epsilon" phrasing implies that this is indeed the case.) If so, then I would argue that both papers have value, just to different audiences.

If I don't need the awesomely frobingent power of the FCT in order to demonstrate that my runcible minacule is both bracticating and cromulent, I may well be perfectly delighted to have the PCT available for direct citation, allowing me to use it with far less comment and/or explanation than if I had to invoke the FCT in its entirety.

• Someone who just wants to apply the theorem directly (and does not care about the proof) would obviously prefer the full theorem, as its statement is shorter, more elegant and easier to understand. However, someone who wants the proof would obviously prefer the partial theorem (if they can get away with it), as its proof is more straightforward. From my (limited) experience, the latter is quite likely to happen. When I had to borrow results from the literature, I was often unable to directly apply them and had to redo the proof (of a suitable variant). Commented Jan 2, 2017 at 23:53