How do you deal with being stuck in mathematics research? With a few other professors, I have been working on proving a certain mathematics conjecture. Our solution seems to be working perfectly in computer simulations but we can't find a proof. (..its a very big result if it works!...) 

I have been trying pretty much everything I can think of. Nothing is working. 

What do you do in such a situation?
A journey of a thousand li begins with a single step.

Rather than aiming for a big result all in one go, why not break the problem down by aiming at some smaller results that will bridge you towards where you want to go.  For example:


If you restrict to a much more limited range (even to a single scenario!), can you show the result for that special case?  You may then be able to generalize bit by bit.
Can you identify some property P such that, if property P holds, the overall result will hold?  You can then separately try to show that your system has P and that P implies the desired result.
Can you break your system into little pieces and take an inductive approach, where you show that if all but one fragment are correct, then the last fragment will be correct too?  You can then show that this holds for larger and larger fragments.


Complementarily, if the simulations work but you can't find the proof, it might be the case that the result does not hold, but it appears to because your simulations are all coincidentally close to some special case.  I have often seen this happen to people.  So an alternate approach that you can take is to try to break your simulation.  For example:


What happens when you hold all simulation parameters constant except for one, and change that parameter greatly, say across a range of a couple of orders of magnitude?
What happens if you radically alter the initial conditions, e.g. making initial conditions extremely homogeneous or heterogeneous in a strongly correlated way?


If the simulation fails in more extreme conditions then, well, you know why you couldn't find a proof.  If it doesn't fail, then you may gain insight into why it doesn't.