Golden Ratio Arctangents Yield New BBP Binary Digit Formulas for π — E8 Intelligence Research

FINDING: Arctangent identities involving the golden ratio yield new BBP-type binary digit extraction formulas, linking φ to π's binary expansion. | MATH: Key result from arXiv:1603.06307 — for odd powers of φ, arctan(1/φ^k) satisfies BBP-type identities. Example: arctan(1/φ) = π/10 (since tan(π/10) = 1/φ). More generally, arctan(1/φ^(2n+1)) can be expressed as rational linear combinations of π and arctan(1/φ^m) with Fibonacci/Lucas coefficients. The BBP formula: π = Σ_{k=0}^∞ (1/16^k)(4/(8k+1) − 2/(8k+4) − 1/(8k+5) − 1/(8k+6)). New formulas: for φ = (1+√5)/2, arctan(1/φ^3) = π/4 − arctan(1/φ) (from Fibonacci identity F_3=2, L_3=4), and binary BBP-type: arctan(1/φ^(2n+1)) = Σ (1/16^k)(...)/(8k+m) with coefficients from Lucas numbers. | CONNECTION: φ itself is the golden ratio (1.618), and its reciprocal φ⁻¹ = 0.618. The arctan(1/φ) = π/10 directly ties the golden ratio to base-10 and to the decagonal (10-fold) symmetry — crystallographically forbidden in periodic crystals but present in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com