Universal Double Balance in Fibonacci Carry Words Across Power-of-Two Bases
Year: 2026
Type: article
Abstract: This paper establishes two universal balance laws for Fibonacci carry words across all power-of-two bases (b = 2^m, m ≥ 1). For complete period-spaced carry words of length b, the binary carry indicators are grouped into nonoverlapping pairs of types 00, 01, 10, and 11. Two signed pair imbalances are defined: Delta = N₁₁ − N₀₀ Eta = N₀₁ − N₁₀ We prove the universal bounds: |Delta| ≤ 2 and |Eta| ≤ 3. These bounds hold independently of the base, digit depth, and starting index. For bases b ≥ 8, the results extend to arbitrary additional lift depths, with exact depth invariance of both imbalances. As the base increases without bound, the carry-word length grows exponentially while both absolute imbalances remain uniformly bounded. Consequently, their normalized values converge uniformly to zero. A further result establishes multiscale coverage: different power-of-two bases can describe the same binary digit cutoff through nested period-spaced carry words. The simultaneous validity of the balance laws imposes additional structural constraints on periodically sampled subwords beyond those obtained from a single base. The paper also establishes complete coverage of Fibonacci indices by period-spaced windows and distinguishes these results from claims about digit normality. All main statements are supported by arithmetic proofs, with supplementary exact computational verification. The results reveal a universal deterministic double-balance structure in Fibonacci carry sequences, combining bounded absolute discrepancies, uniform asymptotic balance, and multiscale arithmetic organization.
Author: Eigler, Dieter
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225341934