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(I'm a math PhD student)

I'm writing a paper and part of my work relies on an old result of my advisor.

Recently, I started to become concerned that the old result does not have a proof (only a sketch) and I have a hunch that the proof is considerably harder (or possibly just much trickier) than the sketch makes it look. (It's possible I'm wrong; it concerns an area that I'm more of a novice in. However, I know enough that I couldn't cite this result without at least reading the sketch.)

Not only that, there are certain questions I have not been able to address as it pertains to my results, and the more I have thought about the missing proof of the old result, the more confident I have become that those questions could be addressed if I knew the missing proof. Therefore, not only does some of my work rely on this old result, but I may be able to substantially improve it if I figure out the old one. (To put it differently, in my own brain at least, the old result is an important open question.)

Further (and, ironically, this was brought to my attention by my advisor), there is a rather big theorem in our research area that relies on the old result. It's troubling to me that I don't know how to prove the big theorem without it, and it could be an issue for the larger community if, as I suspect is possible, no one knows how to fill out the sketch in the original paper. It may be that this is simply an open problem that has been overlooked.

I am discussing this with my advisor and I'm concerned that he won't be willing to show me how to fix the sketch or to make time for us to come up with our own proof. (We discussed it previously, before I had understood its importance, and his explanation followed the sketch in the paper. It looks possible that my advisor and his coauthor overlooked the difficulty in turning the sketch into a proof.)

What's worse is my advisor is starting to get a bit defensive about it (or that's my impression). I don't doubt that he will eventually hear my concerns, I'm simply worried that he will want to move on or will admit that he no longer knows how to fix it; and I don't want to sit around waiting until that happens.

Is now a bad time to reach out to a junior person in the field (who I know and trust) asking if they know how to turn the sketch into a proof? I would not phrase it as "I'm concerned the old result is wrong," but as "What am I missing? I'm new to this particular problem, is it known how to address this kind of issue? Have you read this part of the paper before?" I would ask that this person not discuss it with my advisor before we had come to a conclusion and have the entire conversation via Zoom rather than e-mail. At the end of the day, I don't want to bother my advisor with technical details from old papers, whereas the person I have in mind is someone I trust to listen to my questions and take the details seriously.

My concern is if my advisor hears about it, he could misunderstand my intentions and think I'm telling others his old result is wrong. However, this issue is pretty distracting right now and I just want to know I'm not missing something that really is obvious or understood by experts. If that's the case, then I can stop pressing my advisor about it and move on to other things.

I could wait until this takes its course with my advisor; it's just that he's busy. In retrospect, I wish that I had asked someone else first (before my advisor) so that I wouldn't be in the awkward position I'm in now.

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    It’s an interesting situation and you seem to have given the issue a considerable amount of thought. However... it’s really unclear what exactly you’re asking. “Is now a bad time [...]?” - that’s the only sentence in the post that ends with a question mark, is that literally your question? I have no idea how to approach giving you advice given such a vaguely formed question. Also your post is too long, I suggest that you edit to boil it down to the essentials and make it clearer what kind of advice you need. – Dan Romik Sep 14 '20 at 23:03
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Going to the younger colleague is fine, but I also wouldn't hesitate to go to your advisor again. It is, after all, part of his job to train you. Prepare a detailed explanation of the problem you see with the proof and the spots where you think it is tricky. If you're conscientious in narrowing down the scope of the issue, you surely won't be wasting his time. Either he can fill it in (and you're done) or he can't (and then you've made him aware that there's a gap in his old proof, which is doubtless important to him).

I have been in a similar situation. There was a theorem which my advisor attributed to another, famous and famously sloppy mathematician who only sketched the proof. Moreover, another student of my advisor several years prior wrote out a detailed proof in his unpublished thesis. I was working on generalizing the result to a different setting (already as a postdoc) and noticed an unproven assumption in the other student's proof. In the end the assumption was false, the overall result was correct, but required additional novel ideas and evinced some unanticipated subtleties -- my advisor and I published a paper in a very good journal. (Then I got distracted onto other projects and never finished the generalization.)

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It's absolutely possible that the theorem is wrong. Even with peer-review, invalid theorems get published, especially when a plausible looking sketch is presented. Equally, it is possible that the theorem is true.

Moving forwards, try to expand the proof sketch into a proof. If you get stuck, try to construct counterexamples. If you can't proceed, reach out to the junior person you mention. You could do so casually, e.g., explain that you are trying to construct the proof as an exercise to improve your skills and to better understand the nuances.

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