Koide Relation at −9 ppm: Q = 2/3 from σ-Equivariant Mass Matrix

Overview

IV.T143 derives the Koide relation Q = 2/3 from the σ-equivariant mass matrix of the lepton sector. The σ-equivariance condition on the charged lepton mass matrix forces Q = 2/3 exactly in the leading-order approximation, with an NLO correction at −9 ppm relative to the exact value. The Koide relation is not a numerical coincidence but a structural consequence of the lepton mass matrix being equivariant under the σ-symmetry (the antipodal symmetry of the lemniscate).

Detail

The Koide relation Q = (m_e + m_μ + m_τ)/(√m_e + √m_μ + √m_τ)² = 2/3 holds empirically to very high precision (measured deviation ~ 10⁻⁵) but has no explanation in the Standard Model — it is regarded as a numerical coincidence. Book IV derives it from first principles. The lepton mass matrix in τ is required to be equivariant under the σ-symmetry, which is the antipodal symmetry of the lemniscate L = S¹ ∨ S¹: σ swaps the two lobes (γ ↔ η equivalently). A σ-equivariant mass matrix must have a specific algebraic structure: the eigenvalues satisfy a relation determined by the symmetry class of the representation. For the three-generation charged lepton representation (IV.T172), the eigenvalue relation forces Q = 2/3 at leading order. The δ = 2/9 NLO correction follows from δ = 2/9 =

lobes

axioms

= 2/9, placing Q at −9 ppm from 2/3.

Result Statement

IV.T143: Q = (m_e + m_μ + m_τ)/(√m_e + √m_μ + √m_τ)² = 2/3 from σ-equivariant mass matrix, at −9 ppm.