Can there be a copyright on negative code?

While in some sense this is really a question for Law SE, the common-sense version of it is a matter of whether there are original ideas in the deletion and whether the changeset is expressible (not necessarily as a patch, just in some comprehensible form) in a form that's insufficiently original/creative to be subject to copyright. As the extreme cases, consider:

• Case 1, pretty clearly no copyright: "Remove froblicate function and all calls to it."

• Case 2, pretty clearly copyright: Imagine a giant useless program consisting of an lexicographic-ordered list of all possible statements of length at most 80 characters in a language, where the "modification" of the "program" is a list of lines to remove to make it into a useful program.

You can imagine an entire spectrum between these. Unless it's incredibly obvious, where a particular case falls is really a matter for a lawyer.

Not all changes are of sufficient novelty to constitute something copyrightable, whether they are additions or deletions.

For a simple example, consider any old out of copyright song or hymn of five verses. I could make an 'arrangement' of only the first, second, and fifth verses, but this should not be considered novel enough to be copyrightable.

So in terms of your question about code, I think we would have to see the actual code and what was deleted to make any judgement about it. If what was removed was previously cleaned encapsulated in module-like code (not necessarily using any programming language's module syntax) and the whole module was removed and nothing else, as a non-lawyer I'd guess it probably doesn't reach the level of being copyrightable. But if the deleted code was thoroughly interspersed through the code which was kept and it required a lot of thought to determine what should be kept and what should be deleted, then I'd guess it probably can be copyrighted. But if it matters to you, get a lawyer!

For a non-code example of the later, consider the idea of taking an old out-of-copyright novel, perhaps Little Women, and deleting any sections which don't pass the Bechdel test. I'd think the resulting work is probably novel enough to be copyrightable, especially if they were careful to leave enough connecting sentences that it still makes sense or has a logical plot.

So if B's contributions are thoroughly intertwined with A's original code, probably the best way forward is not to try to cut out only B's code, but all the affected sections or modules even though parts of them were authored by A not B. Then C can replace the whole modules and you don't need to worry about B's contributions any more.

I believe the question makes the assumption that there's just a single work and copyright can either apply or not.
However, we have two works which are separately copyrightable: The commit history, and the code at a specific version.

You get copyright in the history if you have a commit there, since you are one of the many authors of that history.
You get copyright in a specific version of the software if you code is in it. If your lines get deleted, you retain the copyright to the versions that have your lines, but not to later version.

If you modify a work by removing from it or making rearrangements, the modifications you have made are based on your ideas which manipulate someone else's expression. Copyright is concerned with expression, not ideas.

The copyright statutes in the United States do not define what "expression" is. If you wanted to assert copyright over something where your only contribution was chopping, rearranging and deleting existing content, you would have convince a judge that it plausibly constitutes an expression, which would probably be difficult.

For one thing, your expression isn't a piece of content whose existence can be demonstrated on its own. You cannot show the judge:

• here is the material I expressed, viewed separately from the combined work in question; and
• here is where that same material appears in the combined work, which should therefore acknowledge me as a copyright holder

The material you expressed, in isolation, is just a blank, a nothing.

The US Copyright Law has a paragraph about derived works, 103.

Fistly, (a) states that if a derived work is unlawful, then it enjoys no copyright protection. This has implications for open source software, because, by default, a work grants no permissions for making derived works. The permitted ways for making derived works are defined by the specific license. If the license happens to say that the author of a derived version who simply deletes or rearranges lines of code is not permitted to add their name to the copyright notice, then that holds. If the license is violated, then it lapses, leaving that person with no permission to redistribute. Most open source licenses neglect to have such wording, however.

Subparagraph (b) is about lawful derivative works; this is more relevant to the present topic, has this sentence: [t]he copyright in a compilation or derivative work extends only to the material contributed by the author of such work, as distinguished from the preexisting material employed in the work, and does not imply any exclusive right in the preexisting material.

A deletion or rearrangement does not look like "material contributed", which is "distinguished from the preexisting material". All portions of the derived work are material which is preexisting. Concretely speaking, no line of code in the material is one which cannot be found in the original version.

Of course, people will try anything in court, and sometimes there are surprises.

For me, this exposes a weakness in the mental model many coders seem to have about the operation of copyright.

Consider a pile of bricks, representing code contributions to a work. In one (surprisingly common) model, each brick is painted in a colour representing its licence status; red for BSD, blue for GPL, green for Apache, and so on. Whoever made and placed any given brick can consent to its repainting, but nobody else can, though anyone can remove a brick from the pile. In this model, a pile of A's bricks, painted blue, is added to with blue bricks from B. A now wishes to paint the pile red, but cannot because of B's bricks, so sie asks C to make some red bricks with which to replace them. Once the pile is entirely composed of A's blue bricks and C's red bricks, A repaints hir bricks red. How can the sometime presence of B's blue bricks be a problem?

According to a practising barrister and sometime lecturer in copyright law with whom I have discussed this¹ a better mental model is a pile of bricks under one or more comparably-coloured tarpaulins (waterproof covering sheets, aka tarps), with names written on them. Here, anyone can add or remove bricks, but any time you touch a pile under one or more tarps, you add your name to the list(s) written on the tarp(s). The colour of a tarp specifies certain rules: for example, bricks removed from under a blue tarp can't be used in any other pile which is not also under a blue tarp. You can in some cases combine piles under various tarps, and throw a new tarp over the whole lot, without necessarily removing the under-tarps. Certain operations on certain coloured tarps (eg, replacement of a blue tarp with a red tarp) requires the consent of everyone whose name is written on the tarp.

I do not mean to suggest everyone should adopt this mental model, and certainly not all the time, because like any abstraction it too has problems. But thinking about it may reveal to you when you're (quite possibly unintentionally) using the coloured-bricks model, because that is not a good model for copyright. For a start, coloured-bricks falls foul of the Ship of Theseus problem, as we have well-documented here. The tarp model has no issues with the Ship of Theseus, and that alone, to my mind, makes it useful.

The tarp model makes understanding this question less-problematic. Firstly, A piles up bricks under a blue tarp with A's name on it. B comes along, removes several, and writes hir name on the tarp. A replaces them. C then slides a small pile of bricks, under their own tiny red tarp, underneath the big blue tarp, removes some of the bricks replaced earlier by A, and adds hir name to the blue tarp. You now have a blue-tarp-covered pile with three names on the tarp: A, B, and C. The consent of all three will be required to replace the blue tarp with a red tarp (again, with all three names on it).

In short: A, B, and C are all rightsholders in the current work, because all have made contributions to it. The consent of all will be required to relicense it away from GPL.

If A can apply C's changes to the last version before B did any work on it, and if C's changes were made without knowledge of anything B had done, then you will have forked the project to a point where A and C are the only rightsholders, and relicensing will be possible without B's consent.

¹Nevertheless, I am an imperfect conduit, and any mistakes are of course mine.