Mathematically? It's a computer algorithm. Its output is deterministic, and both reproducible and traceable.
Give the AI two copies of its training dataset, one with the copyrighted work, one without it. Now give it the same prompt and compare the outputs.
The difference is the contribution of the copyrighted work.
You mention Harry Potter. In Warner Bros. Entertainment, Inc. v. RDR Books, Warner Brothers lawyers argued that a reference encyclopedia for the Harry Potter literary universe was a derivative work. The court disagreed, on the argument that the human authors of the reference book had to perform significant creative work in extracting, summarizing, indexing and organizing the information from JK Rowling's original works.
I wonder if the court would use the same reasoning to defend the work of an AI?