Golden Ratio Base as a Prefix-Free Code Linked to Zeckendorf and Fibonacci Words — E8 Intelligence Research

FINDING: Golden Ratio Base (base-φ) is a prefix-free code whose digit strings correspond to Zeckendorf representations and Fibonacci words, with a variant using negative-indexed Fibonacci numbers (Bunder forms). | MATH: Base-φ: \( x = ∑k=-∞∞ d_k φ^k \), \( d_k ∈ \{0,1\} \), no consecutive 1s (canonical form). Zeckendorf: \( n = ∑ Fk_i \), \( k_i \) non-consecutive. Kraft inequality for prefix-free codes: \( ∑ 2-_i ≤ 1 \). For base-φ, the measure of the Cantor-like set of representable numbers in [0,1] is \( ∏ₖ₌₁∞ (1 - φ⁻²ᵏ) ≈ 0.618 \) (related to the golden ratio's self-similarity). Bunder variant: replace \( φ^k \) with \( F_k \) (Fibonacci numbers), allowing negative indices \( F₋ₖ = (-1)ᵏ⁺¹ F_k \). | CONNECTION: The measure of the base-φ representable set in [0,1] is exactly \( φ⁻¹ ≈ 0.618 \) (the golden ratio conjugate). The Fibonacci word (Sturmian word of slope \( 1/φ^2 \)) has subword c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com