Golden Ratio Arctangents Yield Novel BBP Formulas for Pi — E8 Intelligence Research

FINDING: Arctangent identities involving the golden ratio yield novel binary BBP-type formulas for π and related constants, linking Fibonacci/Lucas numbers to digit-extraction algorithms. | MATH: Let φ = (1+√5)/2 ≈ 1.6180339887. The paper (arXiv:1603.06307) derives arctan identities of the form: arctan(1/φ^(2k+1)) = Σₙ₌₀^∞ (-1)^n / ((2n+1)·φ^(2k+1)(2n+1)) — but more critically, BBP-type formulas: π = Σₖ₌₀^∞ (1/16^k) · [ A/(8k+1) + B/(8k+2) + C/(8k+3) + D/(8k+4) + E/(8k+5) + F/(8k+6) + G/(8k+7) ] where coefficients A–G are rational functions of Fibonacci and Lucas numbers (e.g., F_n = (φ^n − (−φ)^−n)/√5, L_n = φ^n + (−φ)^−n). Specific new identities: arctan(1/φ) = π/4 − arctan(1/φ^3) (trivial), but nontrivial: arctan(1/φ^5) = arctan(1/2) − arctan(1/φ^3) + arctan(1/φ^7) — derived via tangent addition. Binary BBP for arctan(1/φ^(2k+1)) uses base 2 (not 16), giving digit extraction in O(n log n) time. | CONNECTION: φ is the root of x² − x − 1 = 0, whose continued fraction [ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com