Golden Ratio and Silver Ratio in Quantum Calculus: A Search Result Review — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are a mix of popular expositions (PBS, Numberphile, Sybermath) and one substantive arXiv paper (2410.04169v2) linking golden/silver ratios to Fibonacci divisors, quantum calculus, and supersymmetric oscillators. No direct CHSH–golden-ratio relation is established in these sources. | MATH: arXiv paper: quantum calculus with two bases qϕ = ϕ = (1+√5)/2 and qσ = 1+√2 (silver ratio); Fibonacci divisor derivative Dq f(x) = [f(qx)−f(x)]/[(q−1)x]; Binet form Fn = (ϕn − (−ϕ)−n)/√5 extended to operator F̂N acting on Fock space; energy spectrum En ∝ ϕn or σn for golden/silver oscillators. Nested radical identity: ϕ = √(1+√(1+√(1+⋯))). | CONNECTION: Golden ratio ϕ ≈ 1.618 and its inverse ϕ−1 ≈ 0.618 appear as base of quantum calculus; silver ratio 1+√2 ≈ 2.414 is related to \( \sqrt{2} \
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 Ratio and Silver Ratio in Quantum Calculus: A Search Result Review — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658269},
url = {https://zenodo.org/records/23658269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio and Silver Ratio in Quantum Calculus: A Search Result Review — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658269
UR - https://zenodo.org/records/23658269
AB - FINDING: The search results are a mix of popular expositions (PBS, Numberphile, Sybermath) and one substantive arXiv paper (2410.04169v2) linking golden/silver ratios to Fibonacci divisors, quantum calculus, and supersymmetric oscillators. No direct CHSH–golden-ratio relation is established in these sources. | MATH: arXiv paper: quantum calculus with two bases q_ϕ = ϕ = (1+√5)/2 and q_σ = 1+√2 (silver ratio); Fibonacci divisor derivative D_q f(x) = [f(qx)−f(x)]/[(q−1)x]; Binet form F_n = (ϕ^n − (−ϕ)^−n)/√5 extended to operator F̂_N acting on Fock space; energy spectrum E_n ∝ ϕ^n or σ^n for golden/silver oscillators. Nested radical identity: ϕ = √(1+√(1+√(1+⋯))). | CONNECTION: Golden ratio ϕ ≈ 1.618 and its inverse ϕ^−1 ≈ 0.618 appear as base of quantum calculus; silver ratio 1+√2 ≈ 2.414 is related to \( \sqrt{2} \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -