My math paper recently got accepted and I was told that I will be sent a proof soon. It's just four lines I want to add.. is it OK to add it to my paper? It's additional explanation of something, and I would tell the editor about it.
But is it commonly done or done at all?
In general no. There may be some circumstance when it can be necessary but the proofing stage is too late for any substantial changes. The manuscript should be published in the form it has been accepted and it is an authors responsibility to provide a final version of the manuscript which ideally should not require any changes once type-set and provided in the form of a proof. Changes that are acceptable are changes that do not in any way change the science of the paper.
If any major changes are done to the paper at proofing, the editor will likely decide whether or not these are permissible. What would happen if anyone tries to push substantial changes altering the core science of the paper probably varies, but in the worst case the paper might have to undergo new review.
It is not uncommon to make some changes to articles while they are in press. However, there are a couple of things you should be aware of. First, you ought to contact the editor, to find out whether there is time to insert anything. After a certain point, the publication schedule will probably not permit any changes. After that point, the editor may be willing to delay your paper until a later issue, or they may not.
The second thing, which is more specific to mathematics, is that the editor may not want to make any change that involves the actual mathematics without sending the article back to the referee(s). Mathematics journals have good reason to be very picky that every symbol and argument that appears in a final publication has been subject to the proper peer review. Additional background or explanatory material will probably be fine, but any change to how you calculate or prove something may not be possible without additional refereeing.
