As a grad student how to approach a possible breakthrough research? I am in this peculiar position whereby I started off my grad school in subject X and got very boring projects in it and eventually around my 3rd year I started studying a new subject Y  (partly because I met some inspiring profs in Y who motivated me) 

Now in some sense it was just lucky that after about 6-7 months of working in this new field I along with another co-grad student hit upon a breakthrough research idea possibly solving a famous long-standing open question in this field Y! It was sheer luck that we hit upon an idea which seems to work! (though we are struggling hard to prove the theorem)

Now when I am trying to officially shift departments/institutes to pursue Y, I am explaining in my statements this new possible breakthrough that we hit upon. But I am a bit worried as to how senior established profs in Y will react seeing such a thing in my statement.  



I think all answers till now are completely missing the point of this question and what I am worried about. So let me break-up the context in parts. 

(1) In mathematical stuff no one wants to put on arxiv a "possible" method unless one has a rigorous proof for it to work. If we had a proof then I wouldn't be asking this question! 

(2) In my SOP if I don't write about this progress then what else do I write about!? This is the main stuff that I have been working on ever since I decided to shift fields. So I don't see a choice but to write about this idea!

(3)  The professors who have seen the idea seem to have no opinion at all. They take the view that unless there is a rigorous proof how do we know whether this is right or not. At this point all evidence is numerical simulations. (and the prof who originally inspired me to start Y remarked that he had himself independently also tried one of the ideas we came up with)  

(4) I have anyway linked from my SOP a draft which explains the idea.
As a good researcher always be cautious with "breakthroughs". I would explain that you think it is a good idea worth pursuing. Coming off too strong may put new professors off. Unfortunately I have witnessed many times people believe they have come across breakthroughs when they are nothing of the sort. I think partly because unlike how science is portrayed in the media, "most" excellent science is a long grind, through years of experiments, thought, collaboration and trial and error. The road is often long and winding, and to approach someone well versed in the field with a "breakthrough" may be hubris. But, if it is a strong idea and worth pursuing with good reasoning/logic/science behind it, I cant see a problem convincing anybody in your field the importance of what you propose! Good luck!