Keep my promise not to work? Short version: a potentially upcoming 5-authors maths paper already have authorship dispute drama. I am not among the 5, but I believe I found the solution to the part they got stuck on. Should I tell them?

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Full story:

A is a new graduate student, never have any papers, decided to work with B a post-doc (who had a few paper before) on a topic. They did some significant amount of work, but still nowhere close to a result. A tells his friend C (another graduate student with 1 paper), not to ask for help but just to let him what kind of thing he is working on. But unexpectedly, C got interested and work on it for a while and get some new results, before telling A and B. The work is significant, but A claims that give it a few more weeks and they could have gotten that themselves. C's advisor come to his defense and insist that it is only fair that C got credits as well. Unfortunately, C asked D - an undergraduate in his junior year - to develop a computer simulation, which ended up allowing C to realize that a tweak to the hypothesis is necessary. C never told D what it is for, but D realized that after chitchat with B. Now D's advisor also want him on the paper, since he did some work, and it would really help him in applying to grad school if the paper can be done soon. Neither A nor B is happy about this of course. To make thing worse, E a post-doc from CS department, happen to see the problem and some work on the blackboard. He knew some papers in CS that happen to solve some aspect of the problem, and told B about it, which eventually cause their earlier work to be much more simplified. B thinks E deserve coauthorship too, much to A's resentment.

Now I was not there for the drama. A told me about this over beer a few months ago. I curiously asked what the problem is, and A told me everything including what they are stuck on, after making me promise to never work on this and never tell anyone else about the problem (A don't want to have a 6-author paper). I kept the promise, until recently, when reading some papers, and I found something and realized that this could be as well the key to the problem. After some more computation, I am 90% sure that this method would solve the problem, but I promised not to work on it, so I did not work out everything. Now I am torn. They are unlikely to ever know about this method (I'm in a very different specialty), but their effort had been walking in circle for a few months now. I had never had a paper before, so if I do put effort into this, at least I wanted to be recognized for it.

EDIT: as far as I know, there are only 10 people who know about the paper, us 6 and 4 professors who don't really care nor contribute; it is not public. I think A considered me his emotional crutch which is why he told me about it. The work is probably not good enough for publication, right now it consists of a bunch of different approaches and reformulation that all seemed very close to solve the problem, but did not.
It sounds to me like you're asking this question in the  hope that someone will come along and tell you to do the right thing. Well, here goes: you need to do the right thing and not break your promise. 

Honestly, aside from the obvious moral and ethical argument (which by itself is already pretty compelling), even from a purely selfish and utilitarian point of view I don't see how anything good can come out of you surreptitiously solving the problem and then claiming your place as the sixth coauthor. Given that apparently this is not going to be a very major result or paper anyway, the only thing you will achieve is to make an already messy and acrimonious situation only more messy and acrimonious. Most likely the paper will never get written, and even if it does, everybody including you will come away with a bad taste and little to show for all their efforts and frustration other than a dubious coauthorship, whose value will be more than negated by the effects on the reputations of a few of the people involved, especially you for breaking an explicit promise.

With that said, I think there are two options that could potentially let you participate in the work and benefit from your idea without breaking any promises or acting dishonestly:


Tell A that you have an idea that may be used to attack the problem. Make it clear that you intend to honor your promise not to work on the problem, unless he gives you explicit permission to do so (until the 5-author paper is made public, at which time your promise becomes void anyway). At the same time, you are also under no obligation to tell him your idea, and may legitimately ask to be allowed to officially join the project (with the implied promise of coauthorship of the paper in the event that you end up making a meaningful contribution) as a condition of revealing your ideas to the group. They can then decide if that's something they're interested in or not - either way you are morally in the clear in my opinion.
In the event that you prefer not to discuss your idea with A and the rest of the group, or if you ask them to join the project and they refuse, you can simply wait until they make their results public, and then (as John Ma proposed in the comments) you will be free to use your methods to improve their results and publish your own paper, assuming your improvement is sufficiently interesting to make for a publishable result.


Finally, I should add that although six-author papers are relatively rare in pure math, there is nothing wrong or shameful about publishing such papers, and if each of the authors can indeed claim to have made a reasonably meaningful contribution, then the paper will still have positive value to everyone's careers. Furthermore, going from 5 to 6 coauthors does not involve such a major dilution of the credit (beyond the level to which it is already diluted), so if you have a genuinely new and useful idea to contribute that could make this a pretty attractive proposition for the existing coauthors.