Fixed Points of the Zeckendorf Balancing Map and their Asymptotic Ratios

We introduce a dynamical system on strings of fixed length n associated with linear recurrence sequences (Fibonacci, Tribonacci, Padovan, Pell, Tetranacci, Slow, Fast, and related families). To guarantee completeness of the greedy expansion for every dominant root α > 1, the digit alphabet is generalized to {0, 1, …, M} with M = ⌈α − 1⌉.Given a string of weight k, the generalized balancing map Φ replaces it by the greedy expansion of the difference between the maximal and minimal configurations of the same weight. Iteration of Φ from arbitrary interior starting strings converges rapidly to a fixed point. We prove that the normalized value of this fixed point tends, as n → ∞, to the unified constant M · α / (α − 1).For every order-2 recurrence the constant simplifies exactly to α + 1.Extensive numerical experiments on seven recurrence families at length n = 20 000 confirm agreement with the theoretical limit to at least ten significant digits. Earlier reports of a “dichotomy” for α > 2 are shown to be artifacts of an incomplete binary alphabet; the complete digit set restores a single attractor.A companion paper studies the closely related spatial residual map Ψ, which produces 2-cycles and admits an exact conservation law on bases whose recurrence coefficients lie in {0, 1}.