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Yeah but that will not prevent the stealing, it will only make the fight easier afterwards.

Does it make sense to start privately and then open the repo after publication? Will the dates be retained? Also, isn't commit history easy to spoof?

That is what arxiv is about. We have been facing the same problem with review processes by before. Nothing all too specific here.

At least in my experience, the issue with ArXiv is that the expectation is that the draft should be already in a good enough state. And polishing plus writing the meat around the main result can take a lot of time

This is true, as preprints are addressing the 'problem' at a later point in time. I guess git is better if you do not want to assign an identifier to an immutable version yet.

If only prompts could also be watermarked.

The session data could be cryptographically signed. Probably easier in an open harness?