This page tries to make the case the AI generated code is not copyrightable. But the US Copyright Office's guidelines very strongly indicate the opposite, and certainly doesn't rule out AI Code being copyrightable:

While assistive uses that enhance human expression do not limit copyright protection, uses where an AI system makes expressive choices require further analysis. This distinction depends on how the system is being used, not on its inherent characteristics

However it also makes this point:

The Office concludes that, given current generally available technology, prompts alone do not provide sufficient human control to make users of an AI system the authors of the output. Prompts essentially function as instructions that convey unprotectible ideas. While highly detailed prompts could contain the user’s desired expressive elements, at present they do not control how the AI system processes them in generating the output

but further:

a human may select or arrange AI-generated material in a sufficiently creative way that ‘the resulting work as a whole constitutes an original work of authorship a human may select or arrange AI-generated material in a sufficiently creative way that ‘the resulting work as a whole constitutes an original work of authorship

and

Similarly, the inclusion of elements of AI-generated content in a larger human-authored work does not affect the copyrightability of the larger human-authored work as a whole. For example, a film that includes AI-generated special effects or background artwork is copyrightable, even if the AI effects and artwork separately are not

Note that this isn't settled law though.

Further, note that the failure to register copyright on AI generated images seems mostly because the person attempting this is trying to register it as owned by the AI not a human.

See https://www.copyright.gov/ai/Copyright-and-Artificial-Intell...

I know we're just supposed to upvote and downvote here, but every now and then there's a comment that feels like it deserves an actual thank you for the effort that went into it.

So thank you, for pulling together exactly the kind of details I've looked for but hadn't been able to find.