The Charged-Lepton Koide Relation as a Unit-Dispersion Constraint
Year: 2026
Type: article
Abstract: We record a scale-free reformulation of the charged-lepton Koide relation. Writing the square-root mass vector ${r_{\ell}} = (\sqrt{m_e}, \sqrt{m_\mu}, \sqrt{m_\tau})$ with mean $\bar r$ and standard deviation $\sigma_r$, the Koide quotient is exactly $Q_\ell = (1 + {\operatorname{CV}}({r_{\ell}})^2)/3$ with ${\operatorname{CV}}({r_{\ell}}) = \sigma_r/\bar r$, so that $Q_\ell = 2/3 \iff {\operatorname{CV}}({r_{\ell}}) = 1 \iff \sigma_r = \bar r$: the square-root masses have unit relative dispersion. This isolates the intrinsic shape content of Koide as a single dimensionless number and separates it cleanly from the mass hierarchy, which is fixed independently by the placement of one generation near a node of the square-root profile. We record two structural consequences for the Cosmochrony fermionic sub-programme. First, the specific cascade ladder $(1, \tfrac12 + u, \tfrac12 - u)$, taken by hypothesis as a normal form of the projected generation levels, has a bounded dispersion, ${\operatorname{CV}} \le 1/\sqrt2$ and $Q \le 1/2$, so it cannot by itself realise the Koide value. Second, ${\operatorname{CV}}({r_{\ell}}) = 1$ is the coefficient of variation of the maximum-entropy positive variable at fixed mean; whether the projection entropy ${S_{\Pi}}$ induces such a variational principle on the square-root mass observable is stated as a sharp, falsifiable open test. This note does not derive the masses: it fixes a target invariant for the fermionic-mass sector.
Author: Beau, Jérôme
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166804138
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