Paper LXV: Quark Masses from Flag Manifold Spectral Geometry

Title: Paper LXV: Quark Masses from Flag Manifold Spectral Geometry

Author: Alexander Novickis (alex.novickis@gmail.com)

The Koide formula for lepton masses fails for quarks by 50-67%. We derive a zero-parameter formula $m_q = m_e \cdot \alpha^{-n_q/D}$ from the spectral geometry of the flag manifold $F_2 = SU(3)/[U(1) \times U(1)]$. The denominator $D = \dim(F_2) + \dim(\mathfrak{su}(3)) + N_{\text{roots}} = 17$ counts one-loop soliton modes. The numerators decompose as $n_{\text{down}} = N_{\text{roots}} \cdot \dim(\text{Sym}^g(\mathbf{3})) + g - \text{rank}$ and $n_{\text{up}} = -(N_{\text{roots}} + \text{rank}) \cdot \dim(\text{Sym}^g(\mathbf{3})) + (2D + N_{\text{roots}}) \cdot g - D$, with every coefficient fixed by $SU(3)$ representation theory. All six quark masses agree with PDG 2024 to $0.1\%$–$10\%$. The up-type sign flip is derived from double-well quantum mechanics with the root angle $\cos(120°) = -1/2$. The Cabibbo angle, corrected for soliton backreaction ($\lambda^2_{\text{eff}} = 2/(1+13\alpha)$), gives $\theta_C = 13.08°$ ($0.35\%$ from PDG). All four CKM Wolfenstein parameters are derived: $|V_{cb}| = 1/24$ ($1.1\%$), $A = 0.814$ ($1.5\%$), CP phase $\delta = \arccos(17/48) = 69.3°$ ($1.3\%$), and $|V_{ub}| = 3.60 \times 10^{-3}$ ($1.1\sigma$) via the Weyl group normalization $R_b = \sin\delta/\sqrt{6}$. The Jarlskog invariant $J = 3.09 \times 10^{-5}$ ($0.6\sigma$) requires no experimental CKM input.

Keywords: physics, quarks, mass, topology, flag manifold, soliton, prediction, QCD

DOI: 10.5281/zenodo.19866199

Series: Paper LXV in the Hopf Soliton Programme