Submitting a crude working paper in a Ph.D. application I'm applying to math Ph.D. programs which usually give the option of uploading a sample of some original work. Very recently I have produced some of my first serious results but I haven't yet had the time to write them up anywhere near the standards of publication (and doing so isn't an option since I only have a few days before the big Dec. 15 deadline). For example I'm missing proofs of lemmas, there is no explanatory text in between the math, etc.

My question is whether I have anything to gain by submitting something like a very rough working paper. I'm not sure if it's essential because my statement of purpose contains a brief account of the results and techniques. Also I expect the professor who advised it will validate that it is good work in his recommendation.
It's been a few years since I've reviewed applications for math grad school, but I don't remember ever looking at "mathematical writing samples" of candidates. (I'm not even sure if this is an option at our school.)  As discussed elsewhere on this site (e.g., https://academia.stackexchange.com/a/67732/19607), actual research outcomes are not typically a major determining factor for admission (though a great letter from someone based on your research experience may be).  Thus, I don't think it will make too much of a difference, provided the rest of your application is solid.

That said, if you do decide you want to upload a writing sample (it could make some impression), my advice is don't upload something unpolished.  That almost surely will not make a good impression.  However, you could write and submit a summary/expose of your paper, which will take less time to write.  If in this case, you should try to have one of your professors look over it if possible.