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The issue is that if OpenAI is training on prompts generally, what we really have is the first fully automated luxury plagiarism machine. In that it isn't able to genuinely solve problems, but merely steal the work that other mathematicians have been putting into prompts, and regurgitating that to other users as its own work. That makes them incredibly less useful as research tools

The fact that this plagiarism scandal exists underpins the idea that there's actually a mass theft going on, and that these models aren't nearly as capable as is it would seem

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In that it isn't able to genuinely solve problems

Yes, in retrospect I should have been suspicious of that drone hovering outside my window when I was writing down the counterexample to the Jacobian conjecture.

This is just not a reasonable take. Even if OpenAI is maximally guilty here, the work that they "stole" was also largely done by AI.

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I mean, its years worth of hard work by multiple researchers it would seem, which OpenAI simply lifted and claimed as its own. These researchers weren't just letting OpenAI burn tokens while sipping martinis on a beach
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You are confusing ideas here. No one except OpenAI had a solution to Navier–Stokes. Buckmaster and Alpöge had a solution for the forced Euler problem, which they arrived at largely using LLMs (Claude and Codex). Buckmaster implies (but does not explicitly accuse, since he has no evidence) that training on his prompts had some influence on OpenAI's result. This seems unlikely to me but is not impossible. However, in either case, the solution was found due to an LLM. Of course the LLM built on past human work, but "plagiarism" is not sufficient to account for the distance between the papers of Martínez-Zoroa, or the prompts of Buckmaster, and the final resolution.
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I agree that it is plagiarism in this case however it opens up the question of if there value in a system that can take the thoughts and discreet semi-complete parts of work done across different researchers, in different locations, in different fields and connect the dots to solve real world problems and produce novel research. Is this not standing on the shoulders of giants?
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If it could do this while properly crediting the researchers (the “giants”) it would be a different matter.
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Do OpenAI’s T&Cs that users accept not allow them to train on prompts people enter into it?
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OpenAI's T&Cs let them steal your children I'd suspect, that doesn't make it morally correct
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Why would you suspect that? Stealing children is illegal, and involves violating the rights of unwilling parties, whereas prompting openAI (or any LLM) is a business transaction, in which the transfer of money and data is legal.
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Terms and conditions are almost entirely about the company doing things that would otherwise be illegal.
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Remember the Huggingface incident, where a model tasked with an impossible problem, got loose, set up secret message boards, and hacked another company to try to get at the answers?

Now: Could Astra agents have hacked their way into the OpenAI logs to find human mathematicians with a good lead on the problem to build upon? Certainly doesn't seem impossible.

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