A Conservation Law for the Spatial Residual Map on Zeckendorf Bases

Abstract

We study a spatial residual map Ψ defined on binary strings of length n associated with linear recurrence sequences. Unlike the standard balancing map, which compares a string to the extremal configurations of its weight, Ψ compares a string to itself: Δ(v) = val(M_{k,n}) − val(v) and Ψ(v) = greedy(Δ(v)).On bases whose recurrence coefficients all lie in {0, 1} (Fibonacci, Tribonacci, Padovan, Tetranacci, Slow), Ψ admits an exact conservation law: every 2-cycle {v₁, v₂} satisfies wt(v₁) = wt(v₂) = k and val(v₁) + val(v₂) = val(M_{k,n})exactly. Consequently the normalized sum of the two values tends to α / (α − 1) whenever the common weight diverges.The threshold α = 2 is tied to a concrete breakdown of full-range binary greedy invertibility: any {0, 1}-coefficient recurrence has dominant root strictly less than 2, while bases with a coefficient ≥ 2 (Pell, Fast) exhibit failure rates of 90–99 % on the interval [0, S_n].Sensitivity experiments show that the earlier “two-branch dichotomy” is an artifact of the reconstruction rule rather than of the balancing construction itself: under lazy or dual reconstruction the attractors lock onto the integers 1 and 2.This note is a companion to the paper “Fixed Points of the Generalized Zeckendorf Balancing Map and their Unified Asymptotic Ratios.”Keywords: Zeckendorf representation, residual maps, 2-cycles, conservation laws, greedy invertibility, linear recurrences, numeration systems