mathematics - I found a nontrivial mistake in my currently-under-review paper. Should I send a correction to the editor?

I submitted the paper to a respectable journal about five months ago. My field is mathematics, and I'm at the postdoc stage of my career.

The proof of one of my theorems is more or less completely wrong. It's not just full of small errors; the approach I use doesn't and can't work. The theorem is actually true, and I have a correct proof now. (I've checked the new proof with senior colleagues--yes, I should have done this before submitting the previous version of the paper, but that's neither here nor there.) For context, this is one of three main results of the paper, but its proof only takes up about two manuscript pages (out of 18 total).

Should I send a corrected version of the paper to the editor, or should I wait for the referee report? For those with experience serving as editors: would an incorrect but correctable proof of a main theorem cause you to reject outright a paper you would otherwise accept, or is this more likely to lead to a recommendation of major revisions? I say correctable because neither the theorem nor the (correct) proof is particularly surprising in the context of the problem, and there's a good chance a referee with experience in the subfield would, upon discovering the mistake, have a rough idea of the correct approach to take. (Which, of course, makes the mistake all the more embarrassing.)

Answer

If the mistake is important, you should send a correction now, rather than waiting for the referee report. (If it's not important in the context of the overall paper, then the right course of action is more debatable, but in your case it sounds like an important enough mistake to require notifying the referee.)

1. 
   

   If you wait, you are wasting the referee's time figuring out something you already know.

   
   


2. 
   
   

   I think it looks better to catch your own mistake than to have it pointed out by the referee.

   
   


3. 
   

   If the referee doesn't notice the mistake and suggests accepting the paper, I imagine the editor will rescind the acceptance after you point out the mistake, or at least suggest further refereeing (since not noticing a completely wrong proof calls into question the reliability of the referee report). On the other hand, if you point it out, the referee may still recommend accepting the revised manuscript.

   
   

For context, this is one of three main results of the paper, but its proof only takes up about two manuscript pages (out of 18 total).

This sounds like an ideal case for pointing out the mistake: it has real importance, but I'd bet the chances of acceptance remain reasonable.