Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed

Abstract

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 $F^2 = α^2 μ_\star$ with $α= \sqrt{3/2} - 1$ and $μ_\star = \sum_\ell m_\ell$; identifying this scale with $\bar{m}^{\rm MS}_s(μ_\star)$ yields a light-quark mass cascade with the ratio $m_d/m_s = 5/2 - \sqrt{6}$. Combined with the leading Gatto-Sartori-Tonin relation, this gives the down-sector Cabibbo seed $\sqrt{m_d/m_s} = \sqrt{3/2} - 1 = 0.22474\ldots$, numerically close to the measured $|V_{us}| = 0.22503 \pm 0.00068$ ($0.4σ$). 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.