Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed
Abstract
Several apparently distinct formulations of the charged-lepton mass relation p2/e12 = 2/3 share a common algebraic object: the ratio of flavor-breaking to flavor-symmetric amplitude components on an internal family space. We show that (i) the relation expressed as a Fourier-mode constraint on Shulga's compact cycle, (ii) the same relation as a Descartes curvature condition in the sense of Kocik, and (iii) the Soddy completion of the resulting circle configuration are related by exact algebra, with no approximations. The Soddy completion defines a companion scale F2 = α2 μ⋆ with α= √(3/2) − 1 and μ⋆ = ∑ℓ mℓ; identifying this scale with m̅MSs(μ⋆) yields a light-quark mass cascade with the ratio md/ms = 5/2 − √6. Combined with the leading Gatto-Sartori-Tonin relation, this gives the down-sector Cabibbo seed √(md/ms) = √(3/2) − 1 = 0.22474…, numerically close to the measured |Vus| = 0.22503 ± 0.00068 (0.4σ). A full CKM evaluation requires the up-sector phase correction, which we defer. The correspondence is testable by future lattice QCD+QED determinations of md/ms; explicit failure modes are listed.
Versions
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BibTeX
@misc{brilliant2026descartes,
title = {{Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed}},
author = {Andrew M. Brilliant},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36960},
eprint = {2610.36960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36960},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed
AU - Andrew M. Brilliant
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36960
UR - https://arxiv.org/abs/2610.36960
AB - Several apparently distinct formulations of the charged-lepton mass relation p_2/e_1^2 = 2/3 share a common algebraic object: the ratio of flavor-breaking to flavor-symmetric amplitude components on an internal family space. We show that (i) the relation expressed as a Fourier-mode constraint on Shulga's compact cycle, (ii) the same relation as a Descartes curvature condition in the sense of Kocik, and (iii) the Soddy completion of the resulting circle configuration are related by exact algebra, with no approximations. The Soddy completion defines a companion scale F^2 = α^2 μ_⋆ with α= √(3/2) − 1 and μ_⋆ = ∑_ℓ m_ℓ; identifying this scale with m̅^MS_s(μ_⋆) yields a light-quark mass cascade with the ratio m_d/m_s = 5/2 − √6. Combined with the leading Gatto-Sartori-Tonin relation, this gives the down-sector Cabibbo seed √(m_d/m_s) = √(3/2) − 1 = 0.22474…, numerically close to the measured |V_us| = 0.22503 ± 0.00068 (0.4σ). A full CKM evaluation requires the up-sector phase correction, which we defer. The correspondence is testable by future lattice QCD+QED determinations of m_d/m_s; explicit failure modes are listed.
ER -