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: \( x = \sum_{k=-\infty}^{\infty} d_k \phi^k \), \( d_k \in \{0,1\} \), no consecutive 1s (canonical form). Zeckendorf: \( n = \sum F_{k_i} \), \( k_i \) non-consecutive. Kraft inequality for prefix-free codes: \( \sum 2^{-\ell_i} \leq 1 \). For base-φ, the measure of the Cantor-like set of representable numbers in [0,1] is \( \prod_{k=1}^{\infty} (1 - \phi^{-2k}) \approx 0.618 \) (related to the golden ratio's self-similarity). Bunder variant: replace \( \phi^k \) 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 \( \phi^{-1} \approx 0.618 \) (the golden ratio conjugate). The Fibonacci word (Sturmian word of slope \( 1/\phi^2 \)) has subword c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com