Gavin Roberts

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 digi...

A Conservation Law for the Spatial Residual Map on Zeckendorf Bases

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...

A Conservation Law for the Spatial Residual Map on Zeckendorf Bases

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...

Balanced Zeckendorf Decomposition (BZD) Implementation

Description: This archive contains the Python code and empirical data for the Balanced Zeckendorf Decomposition (BZD), a novel signed-digit Fibonacci representation. Unlike Alpert's (2009) method, which uses rigid gap rules that force index inflation, BZD uses a 3-tier optimizati...