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I sent an article to Annals of Mathematics and after 4 months of review process it was rejected. I received a report from an anonymous expert. Could I send this report, together with the manuscript, to another journal?

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Could I send this report, together with the manuscript, to another journal?

This is a confusing wording: the first two times I read it, I thought you meant you were submitting the referee report for consideration of publication! But I think I understand what you mean now: I'm guessing (though please confirm) that the referee said some good things and in particular regarded, either implicitly or explicitly, the result as being correct, but just not good enough for publication in Annals (as are at least 99.9% of mathematical results), and you would like the positive aspects of the referee report to follow you to your next submission.

If all of the above is right, the answer is that you cannot do this directly: it is simply not the part of the established protocol for paper submission. And there are good reasons for this. On the one hand, how does the next editor know that you did not fabricate the report or modify it in some way? On the other hand, referee reports are never anonymous to the editors, so "submitting an anonymous referee report" undercuts most of its value. However the next journal could contact the Annals and get the report officially and the identity of the referee. Which brings me to what you should do: if you want a referee report from Journal A to be carried over to Journal B, you should write to the editors at both journals and request this to happen. This definitely happens sometimes, although in my experience it is most commonly initiated by someone other than the author.

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I see no reason why the referee report could not be submitted as part of the cover letter, or as an ancillary document. The question to me is not whether it could be submitted, but whether it should. But the question does not contain enough information about the content of the reviewer report to make any suggestion on this. – Wolfgang Bangerth Mar 20 at 17:35
    
I managed to get this done once: the handling editor for the rejecting journal worked at same institution as one that was a realistic 2nd try, and (based on a friend's suggestion) emailed the 1st ed. to say "would you object if I passed your name on?" then added a note to the 2nd submission saying "this paper was rejected from X but they are happy to pass on the referee report if you wish to use it" – Yemon Choi Apr 19 at 2:35
    
Side point: in view of your second sentence, I suggest chasing the link to MSE to see the history of the post and its author. – Yemon Choi Apr 19 at 2:38

Could I send this report, together with the manuscript, to another journal?

I have several times thought of doing just that, and one time decided to take a chance and try it. Somewhat to my surprise, the strategy worked brilliantly: the editor of the second journal decided to accept my submission on the spot without sending it out for another review. Looking back, I think that the forces in play that made the strategy successful (which is also why at the time I decided to try it) were:

  1. The editor of the second journal I submitted to was already somewhat familiar with the work in question and was someone I had a relatively high level of confidence would give my paper a fair hearing.

  2. The referee report from the first journal was fairly detailed, actually said several positive things about my paper, and did not question that the results of the paper were correct.

  3. There was only one particular issue that led that referee to recommend rejection, and that was a somewhat unfair judgment (in my view, which I thought the editor of the second journal would agree with, which he did) regarding the novelty and importance of the paper's results.

Note that this is just an anecdote and offers little predictive value about what would happen in your situation. However, it demonstrates that unusual strategies of this sort can sometimes pay off and that there is nothing that makes them invalid or taboo.

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Did you just include the referee report in the cover letter, or was there some way for the editor of the second journal to learn the identity of the referee? – Arno Mar 20 at 18:38
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As far as I recall it was an email submission, and yes, I did just include the referee report, but I was also fully transparent about where it came from, so the editor could easily contact the first journal to verify the authenticity of the report (whether he could find out the identity of the referee might depend on the policies of the journal). In any case, I don't think he bothered, and those details were largely irrelevant since there was a high level of trust involved - my reputation was good enough and the referee report detailed enough so that those concerns didn't exist. – Dan Romik Mar 20 at 18:51

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