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Any authentic timestamp establishes priority. One can always question the authenticity of timestamps. The arXiv timestamps are perhaps better recognized than MO timestamps. MO timestamps offer the same type of support as arXiv timestamps since MO provides full revision history to everyone and access to the actual database is controlled. (Only one moderator has enough access to potentially alter timestamps.) Both are not fully tamper proof, but the arXiv has a much longer history of such usage. If anyone has thoughts on how to make MO timestamps more trustworthy or reliable, we would be happy to look into implementation possibilities.
How is this a social question?
In my opinion, yes, an MO answer (or any other public online posting) can establish priority. If it were me I would sketch the proof and say explicitly that the paper/preprint will appear soon.
I still tend to think you're more likely to get scooped by being secretive than by being open. If this is something that many people have the tools to work out, the longer you wait to announce, the more likely that someone else will do it.
@Chris Godsil (and Donu Arapura): I am very puzzled by your opinion. Suppose I give a seminar talk wherein I sketch the proof of new Theorem A. I include enough of the key details so that dozens of experts in the relevant field would be able to easily fill in the gaps. Word of this result gets around and two months later, before I've gotten around to finishing my paper, Mathematician X publishes a detailed proof of Theorem A along exactly the lines that I described in my seminar talk. Surely you are not saying Mathematician X (who perhaps was in the audience of the talk) has priority for Theorem A?
If we replace the seminar talk with an MO answer in the above hypothetical situation, I don't think that very much changes.
Maybe you were imagining a slightly different situation, where the sketch in my seminar talk or MO answer fell well short of including all the key ideas. There is a continuum of possibilities interpolating between offering no proof at all and providing an extremely detailed sketch, and I agree that somewhere within that continuum is a grey area where it's not clear that I should get credit for Theorem A. But I think it is common, in situations like this, for it to be possible to give all important details of a proof in a few sentences. In rare cases, the mere statement of Theorem A would be sufficient for experts to easily figure out the proof.
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