So can you write GPL content using Claude? MIT? Because if this condition applies to the output, I don't see how it's compatible with FOSS.

MIT doesn't bar additional restrictions, so you could say this is MIT except you can't train on it. GPL on the other hand does not allow additional restrictions and therefore the code would not be compatible.

but you can't copyright the output anyways. which either makes the restriction on training void or, it means the owners of the model own the copyright, and they only transfer some of the ownership to you. is there such an ownership transfer statement? i haven't seen one yet.

It might depend on your jurisdiction, but LLM output is usually not copyrightable because it is not made by a human. There might be a few exceptions, if you guided the LLM in very specific and particular ways. But in general, the output of an LLM is public domain the moment it was produced.

Can you never tell anybody and just slap a license on it? Sure.

The issue here is not the license, but that you violate their ToS (if that is valid and enforceable is a different question of course). But if you publish the LLM output on github, and someone else takes it to train their LLM, and you did not actively encourage or help them, it's fine.

Yes, obviously, you own the output. Training on those outputs is a violation of the service agreement, and that’s separate from who owns the outputs.