Mediant Trees, Farey Sequences, and Rational Enumeration via Continued Fractions — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Stern-Brocot tree and Farey sequences form a complete, ordered enumeration of rationals via mediants, with deep links to continued fractions and uniform distribution under ergodic weighting. MATH: - Mediant operation: (a/b) ⊕ (c/d) = (a+c)/(b+d) — generates all reduced fractions exactly once in Stern-Brocot tree. - Farey sequence F_n: all reduced fractions between 0 and 1 with denominator ≤ n, ordered; adjacent fractions satisfy bc − ad = 1 (determinant condition). - Continued fraction map: rationals ↔ finite CFs; Stern-Brocot path encodes CF digits (L/R moves). - Uniform distribution result (arXiv:1009.1823): even Stern-Brocot and Farey sequences are uniformly distributed mod 1 under canonical weightings; asymptotic for Lebesgue measure of CF sum-level sets derived. CONNECTION: - The determinant condition bc − ad = 1 is the symplectic/lattice unimodular condition — directly links to SL(2,Z) and modular group, whose fundamental domain tessellates the hyperbol
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026mediant,
title = {{Mediant Trees, Farey Sequences, and Rational Enumeration via Continued Fractions — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635623},
url = {https://zenodo.org/records/23635623},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mediant Trees, Farey Sequences, and Rational Enumeration via Continued Fractions — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635623
UR - https://zenodo.org/records/23635623
AB - FINDING: The Stern-Brocot tree and Farey sequences form a complete, ordered enumeration of rationals via mediants, with deep links to continued fractions and uniform distribution under ergodic weighting. MATH: - Mediant operation: (a/b) ⊕ (c/d) = (a+c)/(b+d) — generates all reduced fractions exactly once in Stern-Brocot tree. - Farey sequence F_n: all reduced fractions between 0 and 1 with denominator ≤ n, ordered; adjacent fractions satisfy bc − ad = 1 (determinant condition). - Continued fraction map: rationals ↔ finite CFs; Stern-Brocot path encodes CF digits (L/R moves). - Uniform distribution result (arXiv:1009.1823): even Stern-Brocot and Farey sequences are uniformly distributed mod 1 under canonical weightings; asymptotic for Lebesgue measure of CF sum-level sets derived. CONNECTION: - The determinant condition bc − ad = 1 is the symplectic/lattice unimodular condition — directly links to SL(2,Z) and modular group, whose fundamental domain tessellates the hyperbol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -