Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle
Abstract
Koide's charged-lepton relation suggests that (√(me),√(mμ),√(mτ)) is the natural family vector. We construct an effective compact-cycle realization in which this vector is sampled from a real amplitude Z(φ) on an internal circle, while the masses arise from quadratic chiral-profile overlaps. The coherent branch-real rank-one sector is an explicit assumption of the effective theory and realizes the equal singlet--doublet norm associated with Koide's 45∘ cone. The empirical proximity of the Z3 phase to 2/9 predates the present work. The new ingredient proposed here is an endpoint-dipole Green response for one elementary family-shift link. The compact harmonic sum fixes the geometric factor γC3=2λB/9, while λB remains an undetermined matching coefficient, and the determinant anisotropy gives θℓ(0)=−2λB/9. Thus λB=1 yields θℓ(0)=−2/9 only as a unit-normalized benchmark, not as an exact first-principles prediction. With the current pole masses, this benchmark requires a relative finite matching correction of 9.83×10−6 in me/mμ; no microscopic theoretical uncertainty for that correction is derived here.
Versions
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BibTeX
@misc{shulga2026chargedlepton,
title = {{Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle}},
author = {Kirill Shulga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.10245},
eprint = {2605.10245},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10245},
}
RIS
TY - UNPB
TI - Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle
AU - Kirill Shulga
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2605.10245
UR - https://arxiv.org/abs/2605.10245
AB - Koide's charged-lepton relation suggests that (√(m_e),√(m_μ),√(m_τ)) is the natural family vector. We construct an effective compact-cycle realization in which this vector is sampled from a real amplitude Z(φ) on an internal circle, while the masses arise from quadratic chiral-profile overlaps. The coherent branch-real rank-one sector is an explicit assumption of the effective theory and realizes the equal singlet--doublet norm associated with Koide's 45^∘ cone. The empirical proximity of the Z_3 phase to 2/9 predates the present work. The new ingredient proposed here is an endpoint-dipole Green response for one elementary family-shift link. The compact harmonic sum fixes the geometric factor γ_C_3=2λ_B/9, while λ_B remains an undetermined matching coefficient, and the determinant anisotropy gives θ_ℓ^(0)=−2λ_B/9. Thus λ_B=1 yields θ_ℓ^(0)=−2/9 only as a unit-normalized benchmark, not as an exact first-principles prediction. With the current pole masses, this benchmark requires a relative finite matching correction of 9.83×10^−6 in m_e/m_μ; no microscopic theoretical uncertainty for that correction is derived here.
ER -