I have changed the name of the project and algorithm because it was potentially misleading or confusing. I first chose it based on the understanding and ideas at the time but they have radically evolved (which is the purpose of these logs, right ?)
As noted before, the "Fibonacci checksum" systems I have found online simply calculate the Fibonacci sequence and obey to the Pisano period related to the word size. It "works" but is not very efficient because the whole state space is not reached. When I added the carry, it changed the whole thing and the algorithm goes from "passable" to "good".
It makes sense that I renamed the algorithm so it is relevant, descriptive and unique.