A bit of an update, too long for a comment.
To be sure we're all on the same page, here's the current setup. If you have sufficient rep, you can click the "close" button on a question, and you get a popup which looks like this:
These are some general close reasons common to the entire Stack Exchange network as I understand, and which, if we wanted to change them, would at the very least require some discussion with Stack Exchange (and they might tell us no).
What we do have control over is the options you get if you choose the second option on this list -- the community-specific reasons. If you do that, you get the following options:
We currently have two options here, and we can have up to three. It would appear that the first option is a generic one we were handed and then never customized further. As a technical point, it appears to be possible for moderators to delete a close reason or to create a new one, but not to edit the description in an existing close reason. We can also tinker with the order they appear in.
As mentioned here, we've just updated the help page to which that first, generic site-specific close reason links:
This question does not appear to be about research mathematics within the scope defined in the help center.
We've now included language there which for the first time explicitly says that these sorts of "check my work" or "result announcement" questions are off-topic. With these updates, the first, generic site-specific close reason technically plays exactly the role suggested by Will Sawin in a comment, because the help page being linked to there now says something about these sorts of close reasons among others. So in some sense this means the answer to the question is "no, we already have a close reason which covers these cases."
However, the close reason in question ("This question does not appear to be about research mathematics within the scope defined in the help center.") leaves something to be desired, because it doesn't summarize what list of reasons it actually covers directly -- you have to follow some link and hunt around a bit to get a list of things this close reason might mean. This is not very helpful either to close-voters or to close-votees.
Therefore, I agree that the close reasons could probably use some kind of update, with a few goals:
The close reasons should be more specific, so that everybody is clear on why exactly a question is being closed.
The specificity of the close reasons should be spelled out in the actual text of the close reason, in addition to containing links to somewhere these reasons are discussed in more detail.
With this in mind, I think it's worth opening up the discussion a bit more broadly. If we have 3 slots for close reasons, I think we can reserve one to be a "catch-all" containing a laundry list of reasons and a link to the help page in case we think to add something later, but we have the opportunity to use the other two in a more specific way. Most likely it's worth keeping the second site-specific close reason (directing the user to Math Stack Exchange) in some form at least, which would leave one more slot. If we have one slot for a more specific close reason, what is the best reason for it to be covering?