Golden Oscillators: Fibonacci Divisors Bridge Quantum Calculus and Supersymmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Quantum calculus with dual bases (Golden and Silver ratios) generates a supersymmetric hierarchy of "golden oscillators" whose Fock-space spectrum is governed by Fibonacci divisors, linking number theory to SUSY quantum mechanics. | MATH: Two-base quantum calculus: q = ϕ = (1+√5)/2 and q = δS = 1+√2 (Silver ratio). Fibonacci divisor derivative ∂ϕ f(n) = [f(ϕ n) − f(ϕ−1 n)]/((ϕ − ϕ−1)n). Binet form for Fibonacci divisor number operator: N̂F = (ϕn̂ − (−ϕ)−n̂)/√5. Energy spectrum for golden oscillator: En ∝ ϕ2n + (−ϕ)−2n (or similar via q-deformed ladder operators), with SUSY partner spectra shifted by ϕ2 or ϕ−2. | CONNECTION: Direct: ϕ = 1.618..., ϕ−1 = 0.618..., ϕ−2 = 0.382..., ϕ2 = 2.618.... The ratio ϕ−1/2 ≈ 0.786 appears naturally in the q-deformed
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Oscillators: Fibonacci Divisors Bridge Quantum Calculus and Supersymmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635215},
url = {https://zenodo.org/records/23635215},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Oscillators: Fibonacci Divisors Bridge Quantum Calculus and Supersymmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635215
UR - https://zenodo.org/records/23635215
AB - FINDING: Quantum calculus with dual bases (Golden and Silver ratios) generates a supersymmetric hierarchy of "golden oscillators" whose Fock-space spectrum is governed by Fibonacci divisors, linking number theory to SUSY quantum mechanics. | MATH: Two-base quantum calculus: q = ϕ = (1+√5)/2 and q = δ_S = 1+√2 (Silver ratio). Fibonacci divisor derivative ∂_ϕ f(n) = [f(ϕ n) − f(ϕ^−1 n)]/((ϕ − ϕ^−1)n). Binet form for Fibonacci divisor number operator: N̂_F = (ϕ^n̂ − (−ϕ)^−n̂)/√5. Energy spectrum for golden oscillator: E_n ∝ ϕ^2n + (−ϕ)^−2n (or similar via q-deformed ladder operators), with SUSY partner spectra shifted by ϕ^2 or ϕ^−2. | CONNECTION: Direct: ϕ = 1.618..., ϕ^−1 = 0.618..., ϕ^−2 = 0.382..., ϕ^2 = 2.618.... The ratio ϕ^−1/2 ≈ 0.786 appears naturally in the q-deformed Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -