Fibonacci Quantum Calculus: Golden Ratio Operators, Arctangent Identities, and Spiral Tilings — E8 Intelligence Research

Abstract

FINDING: Quantum calculus with golden/silver ratio bases yields Fibonacci divisor operators in Fock space, linking supersymmetric oscillators to Fibonacci energy spectra; BBP-type arctangent identities for golden ratio powers; combinatorial tiling with Fibonacci-squared counts; Fibonacci–Theodorus spiral from concatenated right triangles with Fibonacci side lengths. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; Silver ratio σ = 1+√2 ≈ 2.414. Quantum calculus bases: q₁ = φ, q₂ = σ. Fibonacci divisor operator: N_F = (φ^n − (−φ)^−n)/√5 acting on Fock states. BBP-type: arctan(φ^k) = Σ (coefficients)/(b^n) for odd k, binary base b=2. Fibonacci-squared tiling: count = F_n² (from half-squares and fence tiles). Fibonacci–Theodorus: triangle sides F_k, F_{k+1}, hypotenuse √(F_k² + F_{k+1}²) = √(F_{2k+1}) (identity). | CONNECTION: φ = 1.618, φ⁻¹ = 0.618, φ² = 2.618, φ⁻² = 0.382 — all appear in the quantum calculus bases and BBP arctangent arguments. The Fibonacci divisor operator's eigenvalues are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com