oops, it looks like i AM overlooking something! i wrote:

even without this condition. indeed, any two such expressions for x can be transformed to one another by a sequence of replacing a_i F_{i+2} by a_i (F_{i+1} + F_i) or vice-versa; the corresponding

this is not true, sorry. wishful thinking (or lack of thinking) on my part. however, i think there might be something here. perhaps it is the case that the expression \sum_{i,j} a_i b_j F_{i+j} can be transformed into Zeckendorf form by a linear combination of those basic transformations. i will think about it before posting further! mike