> on the other hand if I buy an harry potter book, scan it by hand, print it on paper then burn the print and sell the ashes, that is clearly not a derivative work.

For this analogy, you have to talk not about selling the ashes, but about asserting copyright of your own on those ashes. Your printout of the book would unquestionably be covered by the original author's copyright. The fire would certainly be judged to have made your pile of ashes a transformative use, and any remaining legible letters in your pile of ashes would be in every way de minimus, and you haven't hurt the market for the original book, so you have an extremely safe fair use defense.

Then a more similar case. I make a oil painting of the weight in milligrams of the book (either using or not the ashes in the process).

My point is that the object of the copyright is neither the physical book nor the bitstream in which the movie is encoded, but the book itself and the movie itself.

For example writing a fanfiction of a movie is not transformative because the plot, story, setting, and characters are usually all copyrighted and a fanfiction (generally) use them as-is.

The physical medium is subject to copyright to the extent it can act as a medium for the copyrighted work.

I am not saying that it is legal to sell those first ashes as a product if marketed with the recipe. So I am also not arguing that hosting hashes of movies with the purpose of facilitating piracy is legal (I actually think it is not), but I do not think the reason has anything to do with it being transformative of not.

(I just now realize that ashes/hashes looked like the set up for a pun... it was actually purely coincidental)

Fanfiction of Calvin and Hobs with Calvin the 45 year old parent of an 8 year old girl could easily be transformative without reusing any plot elements. However, if 10 people are all going to end up with the same Hash from the same bit stream you can’t argue it’s a creative endeavor. It’s also not a tiny excerpt as changing even a single bit of the work out result in a different hash. Thus the size of the hash is irrelevant, it’s still copying from the full work.

The only possibility I see is to argue the hash as a simple fact without copyright protection, But, the contents of a book is also a piece of factual information yet copyright is not irrelevant.

PS: As much as you want to use burning as an analogy, a burned book can’t be used to verify a download of the original work where popcorn time’s hashes can.

The hash matches an infinite subset of infinite works. The same thing can glibly be said of any digital representation but the difference is that an algorithm exists to turn the number represented by the flac file back into sound that human beings can enjoy. None such exists for a hash it can never be turned back into the original nor even a piece of same.

Copyright isn't about mathematical derivatives its about protecting the licensed distributor as the sole source of the work. A single screen of a movie or a single page at least is part of the work. Even the first letter of the first word of a book represents a larger portion of the work. It's one nothingth of the work because you can't even derive that first letter from it. Judges are probably substantially interested in the purpose of the law. In math not so much.

> used to verify a download of the original work

I have nothing against arguing that hosting hashes is illegal. It is not that far from facilitating piracy. What I am quite sure is that the hash itself is not covered by copyright.

If you can prove that I am hosting a file whose hash is equal to the hash of a mp4 file that encodes your movie then you can (probabilistically) prove that I am hosting that same file. This is also exactly how bittorrent authentication work if I remember correctly. (my only defense would be to exhibit a non-your-movie- file that still has the same hash)

But this would be no different (or at least comparable) than offering an easy to use list of download links. It would still have nothing to do with whether those links/hashes are derivative or not.