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"using the same approach that Buckmaster and Alpoge had been exploring" is imo mealy wording: it seems fairly likely that OA heard Buckmaster and Alpoge were close to a breakthrough, and decided to use their unlimited compute to quickly prompt based on their assumptions about B&As work.
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Zero data retention, wink.

No looksies, wink.

No trainsies, wink.

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Well, Buckmaster says both his and Alpöge's use of Codex was non-institutional, and OpenAI claims the right to train their models on inputs and outputs of non-enterprise users in their service policies [0]. So I'm not sure they were even promised that.

[0] https://openai.com/policies/how-your-data-is-used-to-improve...

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It isn't relevant whether they were promised that. Indeed I think the assumption must be that they were not promised that, since otherwise the author asking if they were would not make much sense.

If OpenAI did use the conversations from Buckmaster and Alpoge, then not disclosing it, explicitly, is plagiarism. If they planned to use that plagiarism to pressure the authors to publish, that is even more unethical. What the terms of use say does not make it any more or less ethical.

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That's absolutely right. Why the downvotes? If OpenAI are using but not acknowledging the work of others that's plagiarism. If they don't know for sure, but aren't performing due dilligence to make sure they aren't, that's also plagiarism.
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Only for ChatGPT, if the user hasn’t opted out. Would mathematicians be using ChatGPT for this kind of work? Genuinely asking, I know nothing about this!
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Yes, mathematicians are. And yes, most of my colleagues did not even know the opt-out was an option.
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If I am reading your question correctly you are asking about chat interface Vs Codex/Claude code? If so, in my experience Codex/Claude code use is widespread for mathematicians who are seriously using these tools.
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