Quantum Calculus Links Fibonacci Divisors to Golden-Ratio Supersymmetric Oscillators — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The only substantive mathematical result is a quantum calculus framework linking Fibonacci divisors to golden-ratio-based supersymmetric oscillators; the rest are popular expositions with no novel mathematical content. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; Silver ratio σ = 1+√2 ≈ 2.414. Quantum calculus with two bases (φ, σ) defines a Fibonacci divisor derivative; Binet formula F_n = (φ^n − (−φ)^−n)/√5 generalizes to a Fibonacci divisor number operator in Fock space. Energy spectrum of "golden oscillators" is hierarchical, built from powers of φ and σ — implying ratios φ^k and σ^k appear as level spacings. | CONNECTION: φ² = φ+1 → 1/φ = φ−1 = 0.618; φ⁻¹ = 0.618, φ⁻² = 0.382, φ⁻³ = 0.236 — these are the classical harmonic ratios. The hierarchy of golden oscillators with φ^n spacing directly encodes the 0.382/0.618/1.618/2.618 progression. Silver ratio σ relates to 1+√2, which appears in octagonal (8-fold) quasi-crystal symmetry — a crystallographic link. The Fock-space str
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026quantum,
title = {{Quantum Calculus Links Fibonacci Divisors to Golden-Ratio Supersymmetric Oscillators — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635592},
url = {https://zenodo.org/records/23635592},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Calculus Links Fibonacci Divisors to Golden-Ratio Supersymmetric Oscillators — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635592
UR - https://zenodo.org/records/23635592
AB - FINDING: The only substantive mathematical result is a quantum calculus framework linking Fibonacci divisors to golden-ratio-based supersymmetric oscillators; the rest are popular expositions with no novel mathematical content. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; Silver ratio σ = 1+√2 ≈ 2.414. Quantum calculus with two bases (φ, σ) defines a Fibonacci divisor derivative; Binet formula F_n = (φ^n − (−φ)^−n)/√5 generalizes to a Fibonacci divisor number operator in Fock space. Energy spectrum of "golden oscillators" is hierarchical, built from powers of φ and σ — implying ratios φ^k and σ^k appear as level spacings. | CONNECTION: φ² = φ+1 → 1/φ = φ−1 = 0.618; φ⁻¹ = 0.618, φ⁻² = 0.382, φ⁻³ = 0.236 — these are the classical harmonic ratios. The hierarchy of golden oscillators with φ^n spacing directly encodes the 0.382/0.618/1.618/2.618 progression. Silver ratio σ relates to 1+√2, which appears in octagonal (8-fold) quasi-crystal symmetry — a crystallographic link. The Fock-space str Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -