Free and Bound Numbers: A Theory of Independence in the Zeckendorf Representation

We introduce a new notion of independence between positive integers grounded in their unique Zeckendorf (Fibonacci base) representation. Two integers are called free with respect to each other if their representations share no common digit position carrying a 1; otherwise they are called bound. We study free decompositions of a positive integer n — ways of writing n as a sum of a free set of Fibonacci numbers — and establish that every integer admits a canonical minimal free decomposition (given by Zeckendorf) and a unique maximal one. For n = Fₖ, we prove that the maximum cardinality of a free decomposition is exactly ⌈k/2⌉. We introduce the splitting operation on free decompositions, show that any two applicable splits commute (local confluence), and deduce via Newman's lemma that the maximal free decomposition of any integer is unique. We further show that the family of free sets forms an independence system but not a matroid on Z>₀, with the obstruction arising precisely from non-Fibonacci integers. Several open questions are stated.