How to deal with an incompetent collaborator
Not all attempted mathematical research succeeds.  Sometimes you work on something for hours, days, weeks...and realize at the end you have too little to show for it, or that you made a crucial error that ruins everything.  This applies in particular to collaborations: just because you get together with someone (or someones) and propose to work on X, it doesn't mean that you will end up writing a paper on X or even making any progress on X.  (This has certainly happened to me.)  

Hows is this relevant to your present situation?  Well, on the one hand you seem worried about disappointing your friend, which I certainly understand, but actually you delivered on the deal: he wanted to collaborate with you, and so you have.  Surely you didn't agree to collaborate until you achieved some specific goal, no matter what, right?  So I don't think you should feel too bad about breaking off the collaboration: all collaborations are parted sooner or later, and most by mathematicians who remain active on other projects.  On the other hand, you use the word incompetent, which seems harsh to me.  Most mathematicians lack the competence to solve (or even to superficially understand!) most mathematical problems, of course.  Just because your friend can't make correct arguments on this topic doesn't mean they are not competent in something else.  Also collaboration gives you a chance to see someone at their worst as well as their best: some mathematicians make a lot of mistakes that no one ever hears about, but people hear about their remarkable successes.  (We generally respect and admire these mathematicians.)  

If you are describing your friend as "incompetent," it sure sounds to me like you want to be out of the collaboration, so I would suggest that you do so, by which I mean ending the research sessions with your friend.  In my opinion the "classy move" would then be to write up what you have so far and ask your friend how they wish to proceed.  They may well decline to be an author on the paper.  Or maybe they will accept to be an author, either because they think they brought more to the table than you do or because professional exigencies make them feel that they cannot turn down an authorship on something they spent this much time working on.  This kind of thing used to bother me.  It doesn't really anymore.  I don't think your friend is going to get tenure "unfairly" because of this one project with you.

Of course once you have decided what to do about your joint work and that you will not collaborate anymore, you are completely free to do further work as you see fit.