Golden Ratio as Extremal Case in Farey Sequence Gap Structure — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22805983
Primary Topic
Advanced Mathematical Theories and Applications
Type
preprint
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preprint

Golden Ratio as Extremal Case in Farey Sequence Gap Structure — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
preprint

Golden Ratio as Extremal Case in Farey Sequence Gap Structure — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Farey sequences and Ford circles encode maximal-spacing rational approximations, with the golden ratio as the extremal case for gap structure. | MATH: Farey sequence \(F_n\): all reduced fractions \(a/b \in [0,1]\) with \(b \le n\), ordered. Adjacent terms satisfy \(bc - ad = 1\) (determinant condition). Maximal gap between consecutive Farey fractions is \(1/n\) (at endpoints), but the *largest minimal gap* across all intervals occurs near \(\phi = (1+\sqrt{5})/2 \approx 1.618\), where the continued fraction \([1;1,1,\dots]\) gives the slowest convergence — hence the largest "hole" in the Stern–Brocot tree. Ford circles: tangent circles of radius \(1/(2q^2)\) at \((p/q, 1/(2q^2))\); tangency iff \(|ps - qr| = 1\). The golden ratio's convergents \(F_{k+1}/F_k\) (Fibonacci ratios) produce the *maximal* gap in the Farey tessellation of the hyperbolic plane. | CONNECTION: Golden ratio \(\phi\) and its inverse \(\phi^{-1} = \phi - 1 = 0.618\) appear as the *unique* real number whos Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
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