Golden-Ratio q-Deformed Oscillators in Supersymmetric Quantum Mechanics — E8 Intelligence Research

FINDING: q-deformed Fibonacci oscillators in supersymmetric quantum mechanics produce golden-ratio-based Fock space spectra, with quantum calculus linking Fibonacci divisors to N=2 supersymmetric hierarchies. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; Silver ratio σ = 1+√2 ≈ 2.414; q-deformed oscillator with q = φ or q = σ; Fibonacci divisor derivative d_F(x) = (x^F − 1)/(x − 1) at x = φ, σ; Binet form F_n = (φ^n − (−φ)^−n)/√5; energy spectrum E_n ∝ [n]_q = (q^n − q^−n)/(q − q^−1) for q = φ, σ; N=2 supersymmetric hierarchy with supercharge Q, Q† satisfying {Q, Q†} = H, [H, Q] = 0, where H is golden oscillator Hamiltonian. | CONNECTION: φ² = φ + 1 ⇒ 1/φ = φ − 1 = 0.618; φ⁻¹ = 0.618; φ⁻² = 0.382; φ⁻³ = 0.236; φ⁻⁴ = 0.146; σ⁻¹ = 0.414 (near 0.382? No — but σ relates to silver mean, Pell numbers). The q-deformation parameter q = φ directly embeds the golden ratio into the Fock space ladder operators: a a† − q a† a = 1. The Fibonacci divisor calculus yields number operators whose eigenvalue Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com