Golden-Ratio BBP Formulas for π² via Fibonacci Arctangent Identities — E8 Intelligence Research

FINDING: BBP-type formulas for π² exist in golden-ratio base, derived via arctangent identities linking Fibonacci/Lucas numbers to odd powers of φ. | MATH: The arXiv paper (1603.06307v1) derives binary BBP-type formulas for arctan(φ^(−k)) for odd k, and golden-ratio-base BBP-type formulas for π². Key identities: φ = (1+√5)/2; Lucas L_n = φ^n + (−φ)^(−n); Fibonacci F_n = (φ^n − (−φ)^(−n))/√5. Arctangent identities: arctan(1/φ) = π/4 − arctan(1/φ³) (classical), but the paper generalizes to arctan(φ^(−m)) sums yielding π and π². BBP form: π² = Σ (1/φ^n) * (A/(n+a) + B/(n+b) + ...) with digit extraction in base φ. | CONNECTION: Direct geometric harmony — φ (1.618), its inverse 0.618, and φ² (2.618) appear as the base and argument exponents. The BBP base-φ digit extraction is a non-integer base analogue of base-60 (sexagesimal) — both are positional systems with irrational/compound bases. The arctangent identities reflect the golden angle (137.5° ≈ 2π/φ²) symmetry in phyllotaxis and quasicr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com