Fixed Points of the Zeckendorf Balancing Map and their Asymptotic Ratios

We introduce a dynamical system on binary strings of length n associated with a linear recurrence sequence (Fibonacci, Tribonacci, Padovan, Pell, Tetranacci, and others). Given a binary string with k ones, we form the difference between the maximal and minimal configurations of weight k and expand that difference greedily in the same basis; this defines a map Φ, the balancing map. Iterating Φ from an arbitrary starting string converges rapidly to a fixed point. We prove that the normalised value of a natural candidate fixed point, val(v)/G(n+1), converges as n→∞ to a constant depending only on the dominant root α>1 of the recurrence: to α/(α−1) when α<2, and to one of α/(α−1) or 1/(α−1) when α>2. We give a complete proof of these ratio limits from the closed form of the recurrence, and we present extensive numerical evidence, independently reproduced for this version of the manuscript, across seven recurrence families and string lengths up to n=20000, confirming convergence to at least ten significant digits. When α>2 the two candidate limits are genuinely distinct fixed-point values; we document numerically that almost every random starting string converges to the low value 1/(α−1), while the specific "balanced" candidate construction itself realises the high value α/(α−1). We formulate a precise conjecture describing this dichotomy in general, discuss its connection to generalized Zeckendorf representations (PLRS/ZLRS) and to Pisot numbers, and record the exact conditions, verified computationally, under which the natural candidate is or is not an exact fixed point of Φ.