Golden Ratio Arctangent Identities Yield New BBP-Type Digit Formulas — E8 Intelligence Research

FINDING: The arXiv paper (1603.06307v1) derives novel BBP-type arctangent identities for odd powers of the golden ratio, linking Fibonacci/Lucas numbers to binary digit extraction formulas, and introduces golden-ratio-base BBP-type formulas. | MATH: Let φ = (1+√5)/2. The paper derives arctan(φ^(−k)) identities for odd k, expressible as sums involving Fibonacci (F_n) and Lucas (L_n) numbers. Specifically, BBP-type formulas of the form Σ (1/16^n) * (polynomial in n) / (denominator involving φ^(2n+1) or similar) yield arctan(φ^(−odd)). Key constants: φ, φ^2 = φ+1, φ^−1 = φ−1 = 0.618..., φ^−2 = 2−φ = 0.382..., φ^−3 = 2φ−3 ≈ 0.236. The arctangent identities exploit the tangent addition formula: tan(arctan(a) + arctan(b)) = (a+b)/(1−ab), with a, b chosen as powers of φ^−1. | CONNECTION: Direct geometric harmony: φ^−1 = 0.618, φ^−2 = 0.382, φ^−3 ≈ 0.236, φ^−5 ≈ 0.090 — all appear as coefficients in the arctangent series. The odd powers of φ relate to the golden angle (2π/φ^2 ≈ 137.5°) and to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com