Isomorphic Rational Trees Link Farey, Stern-Brocot, and Markov Spectra — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Stern-Brocot tree, Farey sequences, Calkin-Wilf tree, and continued fractions are isomorphic representations of ℚ⁺, with the Lagrange spectrum and Markov triples emerging as the arithmetic shadow of badly-approximable irrationals — the same objects that generate the golden ratio and its metallic-mean family.
MATH:
Core structures (all bijective on ℚ⁺): - Stern-Brocot: mediant insertion. From 0/1, 1/0, insert (a+c)/(b+d). Every positive rational appears exactly once. - Farey F_n: fractions in [0,1] with denominator ≤ n; consecutive terms a/b, c/d satisfy bc − ad = 1. - Calkin-Wilf: breadth-first enumeration via f(a/b) = a/(a+b), (a+b)/b. - Continued fractions: [a₀; a₁, a₂, …] ↔ Stern-Brocot path (L/R runs of length aᵢ).
Unimodular invariance: - det[[a, c],[b, d]] = ad − bc = ±1 for Farey neighbors → SL(2,ℤ) action. - Stern-Brocot = Cayley graph of PSL(2,ℤ) with generators L, R.
Lagrange spectrum: - L(α) = liminf_{q→∞} q‖qα‖. - Lagrange's theorem: α badly approximable ⟺
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026isomorphic,
title = {{Isomorphic Rational Trees Link Farey, Stern-Brocot, and Markov Spectra — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658699},
url = {https://zenodo.org/records/23658699},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isomorphic Rational Trees Link Farey, Stern-Brocot, and Markov Spectra — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658699
UR - https://zenodo.org/records/23658699
AB - FINDING: The Stern-Brocot tree, Farey sequences, Calkin-Wilf tree, and continued fractions are isomorphic representations of ℚ⁺, with the Lagrange spectrum and Markov triples emerging as the arithmetic shadow of badly-approximable irrationals — the same objects that generate the golden ratio and its metallic-mean family. MATH: Core structures (all bijective on ℚ⁺): - Stern-Brocot: mediant insertion. From 0/1, 1/0, insert (a+c)/(b+d). Every positive rational appears exactly once. - Farey F_n: fractions in [0,1] with denominator ≤ n; consecutive terms a/b, c/d satisfy bc − ad = 1. - Calkin-Wilf: breadth-first enumeration via f(a/b) = a/(a+b), (a+b)/b. - Continued fractions: [a₀; a₁, a₂, …] ↔ Stern-Brocot path (L/R runs of length aᵢ). Unimodular invariance: - det[[a, c],[b, d]] = ad − bc = ±1 for Farey neighbors → SL(2,ℤ) action. - Stern-Brocot = Cayley graph of PSL(2,ℤ) with generators L, R. Lagrange spectrum: - L(α) = liminf_{q→∞} q‖qα‖. - Lagrange's theorem: α badly approximable ⟺ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -