I feel like you're making a logical leap here by assuming lossy and failure to reproduce in entirety implies inability to recognize. As a trivial example, I can take a sha256 hash of your comment here, lose the ability to reproduce it, but still have an extremely accurate ability to recognize whether some text is exactly your comment or not. Obviously hashing every substring would not be a particularly efficient strategy, but my point is that saying "it's lossy" isn't particularly compelling without other details.

I haven't been following it well but isn't part of the NYT lawsuit against OpenAI that it sometimes spits out NYT articles verbatim?

It is lossy, but it is still enough for verbatim recreations. All of Wikipedia is just 24GB of lossless compressed text and all of JK Rowling's work fits into a few MB. So these things would easily be storable verbatim in trillion parameter models. Reasoning about the training cutoff is also something that the newest models do pretty well, because you can teach them to do so after pre training using e.g. SFT. With tool use it can then even check actual current sources, which may happen without you even knowing in the normal chat apps unless you use a controlled API call.

How do you know, how the model works? If there was an index of all Micken's writings, or even if the model searched the web before feeding the response to you, you wouldn't know by observing from the outside.

that's in the ideal scenario where it's only seen a single copy of it tho

Haven’t there been repeated experiments that show if you jailbreak most frontier models’ harnesses you can get them to output near verbatim copyrighted works?

I swear there was a whole court case about this in the last year.