Golden and Silver Ratios in Supersymmetric Oscillator Hierarchies and Fermion–Boson Entanglement — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 28, 2026• 0 citations

Abstract

FINDING: The only substantive result in this set is a quantum-calculus construction linking Golden/Silver ratios to supersymmetric oscillator hierarchies and fermion–boson entanglement; the CHSH–√2–φ connection is not present in the retrieved sources. | MATH: φ = (1+√5)/2 = 1.6180339887…, satisfying φ² = φ + 1 and φ = √(1+√(1+√(1+…))); silver ratio δ_S = 1+√2 = 2.4142135623…, satisfying δ_S² = 2δ_S + 1; Binet form F_n = (φⁿ − ψⁿ)/√5 with ψ = −1/φ = −0.6180339887…; two-base quantum calculus with bases q = φ and q = δ_S acting on Fibonacci-divisor number operator in Fock space. | CONNECTION: φ and δ_S are the positive roots of x² − x − 1 = 0 and x² − 2x − 1 = 0 respectively — i.e., the two simplest quadratic irrationals of the form x² − nx − 1 = 0 (n = 1, 2). These are the eigenvalues of the SL(2,ℤ) matrices [[0,1],[1,n]], generators of the modular group; φ corresponds to the pentagonal/icosahedral root system H₃ (5-fold symmetry), δ_S to the octagonal/√2 lattice (D₄-related, 8-fold). Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23724157

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