My collaborator has asked to be only acknowledged in a paper. Should I make him a co-author? I am writing a paper in mathematics. It relies heavily on a result from a different topic that I phrased, felt that it should be true, but I did not know how to prove that result. I asked someone who works on this different topic to help me with the proof, and he has found a proof for the desired result. Let’s call him Bob.

I would like to thank this person and ask him if he wants to:


be a co-author in the paper,
just be acknowledged in the paper:


  The author wishes to thank Bob for generously allowing him to use his proof of [result].


At the beginning of the proof of the result it will be mentioned that:


  The proof is due to Bob.



Bob says that he would be happy to be acknowledged and does not think that he should be a co-author (unless he can further contribute to the paper). He also mentions that he is not working on my topic and hence not enough familiar with what the paper deals with.

My question is: Which of the two options (1 or 2) should be chosen?
It should be emphasized that the result I have been helped with is critical for the existence of the paper. Moreover, is it okay to submit the paper to a journal as a single author, and ask the editor to have the opinion of the referees on whether Bob should be a co-author or not?

I read Should all authors on a paper be comfortable explaining every aspect of the paper?, I guess that there are differences between mathematics and other topics, aren't they? For example, in brain research people from different topics collaborate and we do not expect that one will understand what exactly the other did, but in a math paper I would expect that every author will understand what his friend did. Or perhaps I am wrong?
The answer depends on how firmly Y has declined authorship.  I don't think it's appropriate to ask the referees to decide on authorship. 


You cannot list Y as an author without his consent.  If he has firmly refused to be an author, that is the end of it.
If Y has simply indicated that acknowledgment is a sufficient form of credit, but left the door open to being an author then it's up to you.  I have responded in this way sometimes when I felt my contribution was sufficient for authorship but I didn't want to step on anybody's toes.  I try to err on the side of being generous with credit, so I would say to him 



  "I feel that your contribution warrants authorship.  Are you willing to be listed as an author?"


If he still says no, then just acknowledge him.  You can include the statement of how critical his part was in the acknowledgment.

In the comments, there is a suggestion to state in the paper that Y has declined authorship.  I don't think this is wise; it needlessly draws attention and may make people speculate as to why.  If I were Y I would not want that statement in the paper.