Golden Ratio Base: A Unified System of Non-Integer Numeration — E8 Intelligence Research

FINDING: Non-integer bases (fractional, irrational, and specifically golden-ratio base φ) form a coherent number system with unique representations, including Zeckendorf and negative-indexed Bunder forms, linking to Fibonacci combinatorics. | MATH: Base φ (golden ratio base): digits 0,1; value = Σ dₖ φᵏ, φ = (1+√5)/2 ≈ 1.6180339887. Key identities: φ² = φ + 1; φ⁻¹ = φ − 1 = 0.6180339887; φ⁻² = 2 − φ ≈ 0.3819660113. Zeckendorf theorem: every positive integer uniquely as sum of non-consecutive Fibonacci numbers (Fₖ). Bunder forms extend to negative indices: F₋ₙ = (−1)ⁿ⁺¹ Fₙ. Non-integer base representation: for base β > 1, digits 0…⌈β⌉−1; for β = φ, digits {0,1} with no consecutive 1s (canonical form). | CONNECTION: Direct geometric harmony: φ⁻¹ = 0.618 (golden ratio conjugate), φ⁻² = 0.382, φ² = 2.618 — all appear as powers of the base. The Zeckendorf representation is a combinatorial shadow of the golden ratio's continued fraction [1;1,1,1,…]. Base-φ expansions encode the same recurren Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com