The Golden Ratio Bridge: Rogers-Ramanujan Continued Fraction and Quintic Solvability — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Rogers-Ramanujan continued fraction (RRCF) provides a modular bridge between the golden ratio conjugate and the general quintic, revealing that degree-5 solvability is encoded in a continued fraction whose special values are algebraic units in the golden field.
MATH: The RRCF is defined as R(q) = q1/5/(1 + q/(1 + q2/(1 + q3/(1 + ⋯)))) For q = e−2π (or modular transformations thereof), R(q) takes values in ℚ(√5). Key special value: R(e−2π) = √((5 − √5)/2) − (√5 − 1)/2 — an algebraic number involving ϕ−1 = (√5−1)/2 ≈ 0.6180339887. Modular transformation: R(−1/q) = (1 − ϕ R(q))/(ϕ + R(q)) where ϕ = (1+√5)/2. The arXiv paper (1510.00068) expresses a root of the general quintic x5 + ax + b = 0 as an algebraic function of R(q) for a specific q determined by a,b, effectiv
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 = {{The Golden Ratio Bridge: Rogers-Ramanujan Continued Fraction and Quintic Solvability — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23410034},
url = {https://zenodo.org/records/23410034},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Golden Ratio Bridge: Rogers-Ramanujan Continued Fraction and Quintic Solvability — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23410034
UR - https://zenodo.org/records/23410034
AB - FINDING: The Rogers-Ramanujan continued fraction (RRCF) provides a modular bridge between the golden ratio conjugate and the general quintic, revealing that degree-5 solvability is encoded in a continued fraction whose special values are algebraic units in the golden field. MATH: The RRCF is defined as R(q) = q^1/5/(1 + q/(1 + q^2/(1 + q^3/(1 + ⋯)))) For q = e^−2π (or modular transformations thereof), R(q) takes values in ℚ(√5). Key special value: R(e^−2π) = √((5 − √5)/2) − (√5 − 1)/2 — an algebraic number involving ϕ^−1 = (√5−1)/2 ≈ 0.6180339887. Modular transformation: R(−1/q) = (1 − ϕ R(q))/(ϕ + R(q)) where ϕ = (1+√5)/2. The arXiv paper (1510.00068) expresses a root of the general quintic x^5 + ax + b = 0 as an algebraic function of R(q) for a specific q determined by a,b, effectiv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -