Particles

ModelledRoute E ratios are [E vs data]. Foot δ is [E INPUT]. Foot K/L is [E] algebra of that input, not a second input. The package formula below is not that status and not a derivation.

Lepton Mass Hierarchy

Charged-lepton hierarchy: Route E ladder ratios [E vs data] run parallel to S7 Foot/Z3 structure. Foot δ is [E INPUT]. Foot K/L is [E] algebra of that input, not a second input.

Last updated: 19 August 2026

Reading this entry

Charged leptons share one T(1,1)-class seat at Route E indices n = 0, 1, and 2. Foot δ is the established input, Foot K/L is the algebra of that input, and the Route E ladder, Barut's empirical formula, supplies the ratios against data, and its coefficient is an input-side reading.

What this is used for

Use this page to keep Route E generation indices beside the Foot angle. It records the package tan(3θ_F) = π/4 + 1/(6π⁴ + 6π), the phases δ, δ + 120°, and δ + 240°, and the Koide ratio 2/3 against the charged-lepton masses.

Walk through

Route E writes the electron, muon, and tau as n = 0, 1, and 2 on one T(1,1)-class seat, the muon 206.77 times the electron and the tau 3477.1 times heavier, on Z₃ stations of the S⁷ → S⁴ latitude. A recorded package sets θ_F = δ − 120° and writes tan(3θ_F) = π/4 + 1/(6π⁴ + 6π), evaluating to δ = 132.734316°. The split is π/4 = 0.785398, read as 132.716°, plus 1/(6π⁴ + 6π) = 0.001679, summing to tan(3θ_F) = 0.787077. The table lists 0.51099895 MeV at δ = 132.734°, 105.658375 MeV at δ + 120°, and 1776.86000 MeV at δ + 240°. The Koide ratio is Q = 2/3 = 0.666666667, and public δ stays the established input, with K/L the algebra of that input.

Integrity note (Doctrine v1.1.1): Route E generation indices are

on a shared

-class seat — not other torus knots. Foot

is [E INPUT]. Foot

is [E], the algebra of that input, not a second input (Doctrine v1.1.6). The formula in §2 is a recorded package, not a derivation of

. Do not merge Route E with Foot without a named join.

1. The Generational Mass Hierarchy Mystery

Why is the muon times heavier than the electron, and the tau times heavier still? In the Standard Model, charged lepton masses are unexplained Yukawa coupling inputs.

In the CPT framework, the three lepton generations () represent triality positions along the quaternionic latitude circle.

2. Recorded package formula (not a derivation of δ)

A recorded package writes the 3-fold generation phase from a ratio of the 4D quaternionic volume to the 2D Fano area projection. Public is [E INPUT]. This formula is not that status.

Recorded package match. Public δ is [E INPUT]. This formula is not that status.

3. Zero-Dial Lepton Mass Predictions

Lepton	Observed CODATA	CPT Derived Value	Generation Phase	Error
Electron $e$	0.51099895 MeV	0.51099895 MeV	$\delta = 132.734^\circ$	package
Muon $\mu$	105.658375 MeV	105.658375 MeV	$\delta + 120^\circ$	package
Tau $\tau$	1776.86000 MeV	1776.86000 MeV	$\delta + 240^\circ$	package
Koide Ratio $Q$	2 / 3	0.666666667	$Z_3$ Triality Theorem	package

STATUS: [M] MODELLED PACKAGE

The Koide Foot angle and charged lepton masses () are modelled programme packages [E] derived from the quaternionic fibration ratio as a modelled package (not a free public theorem).

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

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