I'm curious. Do questions on math stack exchange have to be about math that has already been documented and can be literarily researched, or can it involve new segments of math that the asker is researching into? I know this sounds strange but I think it's a matter of semantics. After all, newer concepts will be confusing and aren't truly answerable unless the person reading can somehow figure something out that others haven't (or haven't bothered to figure out).
[C]an [questions] involve new segments of math that the asker is researching into?
I personally think such questions are perfectly on-topic.
As Hardmath points on in a comment, it's quite possible that somebody in this wide world of ours has already breached the topic (or something similar, or at least relevant), and you're just not aware. This is great for everyone, because you benefit from their knowledge, and they get to use it for once!
But there are definitely some things to take into consideration.
It still needs to follow all the general guidelines for being a good, or at least acceptable, question.
So among other things, it needs to be readable and understandable (at least in theory). I can think of some examples that were eventually closed because nobody understood what the person was talking about. There was even a significant period in which the OP tried to clarify the question in response to comments, but to no avail.
Math.SE isn't a personal blog. While some people may, a significant portion of users don't particularly appreciate what appear to be personal memos (especially if they're edited heavily over a significant timespan), or long strings of interconnected posts that seem to be relevant and understandable to the poster only.
It could be more likely to be perceived as the kind of thing a crank would ask. There are definitely some cranky questions, more often than not focused on Collatz etc. conjectures (Brocard is also pretty popular), or attempts to invent some new kind of wheel.
It's quite possible that, even if it is a good question that's more or less accessible by a few people, it will go largely ignored. One of my favorite answers is in a pretty obscure and relatively new branch of geometric combinatorics. The question was several months old if not older, and I thought my answer -- the first answer or comment -- was pretty nice. I will be surprised if it ever gets its first upvote $\ddot \smile$
Obviously these concerns aren't unique to questions on ground-breaking territory, nor will you necessarily come anywhere close to having these issues. But I think they're significantly more likely, compared to the usual sorts of questions people ask.
So, ask away! But be prepared to work 15 times as hard as you would to ask a normal question.
I have seen many "out there" questions get heavily downvoted, which one might think is indicative of the math.stackexchange community not liking such questions. But from what I have seen, many of these questions are poorly written and the ideas behind the post might not make sense. Also, the mathematics may not appear interesting, or at least the author hasn't done a good job of convincing people that their ideas are worth investigating. Many times it's clear that the author has not put in any effort into the question themselves.
The problem with introducing previously unexplored concepts is that most people who do math, even recreationally, already have their hands full with definitions, notations, and concepts they are trying to understand. Why should users put in the effort in trying to get comfortable with new, unmotivated concepts that are very much unrelated to the problems they are already putting so much time into?
There is nothing inherently bad about asking a very original, abstract, or isolated question, but the responsibility is on you to convince your audience that the math you're trying to do is actually interesting.