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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.

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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.

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Based on your description of the paper, I suspect it is too rough for it to be a good idea to include it in your application. Math graduate committees will probably be using the paper more as an indication of your overall level of mathematical knowledge, sophistication and skill set than to evaluate the specific results obtained. To say it for the hundredth time: in pure math, one can get into the very top graduate programs without having proved any theorems of one's own. So your results are not going to get measured against the results of most other candidates, because they probably won't have any. However they will get measured on how much mastery you have of the material included there and how much mastery you have of mathematical language in general, and a very rough draft can work against both of those points.

In general, mathematicians applying for their next position do not include rough drafts of their work -- they usually only include (or link to) completed work. Exceptions do occur: for instance, when soliciting recommendation letters I have sometimes sent along a draft of a paper that was not yet publicly available, and if it were really complete it would probably have been publicly available. But (i) this in in a context where my specific work certainly was being evaluated and weighed, (ii) I sent it to someone who would understand and appreciate it much better than a random committee of faculty doing graduate admission and (iii) still I sent a complete draft, rather than one missing explanatory text or minor proofs.

In your situation you really do not need to send the incomplete work, because you have a recommendation letter from the professor who advised you. This is much of what these recommendation letters are for, by the way: they tell of the work that the candidate has just done or is just about to do but has not yet properly written up, they describe it in a more high level way (ideally taking advantage of their broader perspective of the field) and rather than including proofs they can rely on their reputation and stature in the field. So getting a letter which speaks well of the work should have most or all of the advantages of sending the draft with none of the disadvantages of having very unpolished work evaluated.

Let me end by saying that a student who has done significant work but without a faculty member to evaluate it and write about it and who has written it up only roughly is in a pretty bad spot. It is only too easy to question whether the work is actually correct and significant, and in the context of evaluating a large stack of applications there may not be enough time to resolve this. Students should try to avoid this situation!

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