Koide Relation Q

τ-Formula

Q = lobes / dim(τ³) = 2/3

Derivation

in agreement with the experimental value $Q^(exp) = 0.666\,661 ± 0.000\,007$ at $-9$ ppm. The rational value $2/3$ arises from the lemniscate structure: $2/3 = lobes / (τ^3)$. The derivation follows four steps:

• Three generations correspond to three winding classes in $π_1(τ^3) ≅ ℤ^3$ (Theorem (thm:ch60-three-gen-rank)).
• Each class has a mass eigenvalue from the $T^2$ mode spectrum.
• The constraint $χ_+ + χ_- = 1$ on the two lobes of $$ forces a democratic matrix structure.
• The democratic matrix has eigenvalue ratio $2:(-1):(-1)$, giving $Q = 2/3$.

The structurally significant parameter controlling the departure of the lepton mass spectrum from the democratic configuration $m_e = m_μ = m_τ$ is

where lobes $= 2$ (the two lobes of $ = S^1 S^1$) and $(τ^3)^2 = 9$. This ratio controls the Koide phase, setting the angular separation of the three mass eigenvalues on the unit circle of the democratic mass matrix.

Source

This prediction is derived in the Numerical Physics Ledger (Chapter 60 — mass-spectrum), Books IV–V of Panta Rhei.