CA9-WP-2026-001 · Cahiers de physique théorique CA9
A topological reading of the Koide relation: Z₃ structure, the √2 amplitude, and a parameter-free electron-mass estimate
Lilian Alexandre Cariou (CA9 SAS) · ORCID 0009-0006-8276-4048
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Texts in this series are not peer reviewed.
Abstract. Koide's relation Q=(m_e+m_μ+m_τ)/(√m_e+√m_μ+√m_τ)² = 2/3 is one of the most precise numerical coincidences in particle physics (3.3 ppm on the 2024-25 masses), yet the Standard Model offers no explanation. We give it a topological reading: the three charged leptons as the three Z₃ generation modes of a single trefoil knot, the √2 amplitude carried by the sense doubling, the phase decomposed as π/12 − ε, and a parameter-free electron-mass estimate to ±3%. A dated prediction on m_τ (Belle II) is attached.
Record
| First published | 23 June 2026 |
|---|---|
| Online version | v3 — 31 August 2026 |
| DOI | 10.5281/zenodo.20817000 |
| Pages | 6 |
| Licence | CC BY 4.0 |
| Languages | English, with a French edition of every document |
How to cite
Cariou, L. A. (2026). A topological reading of the Koide relation: Z₃ structure, the √2 amplitude, and a parameter-free electron-mass estimate. Cahiers de physique théorique CA9, CA9-WP-2026-001-EN. doi:10.5281/zenodo.20817000
BibTeX record
@techreport{cariou2026001,
author = {Cariou, Lilian Alexandre},
title = {The Koide relation from the Z3 symmetry of the trefoil: Q=2/3 with r=sqrt(2) carried by the sense doubling (conditional), Koide's 1981 relation for m_tau, the phase decomposed as pi/12 - epsilon, and m_e computed to ±3% (TQNT)},
institution = {CA9 SAS},
type = {Cahiers de physique théorique CA9},
number = {CA9-WP-2026-001-EN},
year = {2026},
doi = {10.5281/zenodo.20817000},
url = {https://www.ca9.fr/cahiers/2026/001/}
}