> If the output of a mathematical model trained on an aggregate of knowledge that contains copyrighted material is derivative and infringing, then ipso facto, all works since the inception of copyright are derivative and infringing.

I'm not saying every LLM output is necessarily infringing, I'm saying that some are, which means the underlying LLM (considered as a work on its own) must be. If you ask a human to come up with some copy for your magazine ad, they might produce something original, or they might produce something that rips off a copyrighted thing they read. That means that the human themselves must contain enough knowledge of the original to be infringing copyright, if the human was a product you could copy and distribute. It doesn't mean that everything the human produces infringes that copyright.

(Also, humans are capable of original thought of their own - after all, humans created those textbooks in the first place - so even if a human produces something that matches something that was in a textbook, they may have produced it independently. Whereas we know the LLM has read pirated copies of all the textbooks, so that defense is not available)