The Quark Sector: Mass Hierarchy, Mixing Angles, and CP Violation from Cascade Geometry in Cosmic Egg Theory

The Quark Sector: Mass Hierarchy, Mixing Angles, and CP Violation from Cascade Geometry in Cosmic Egg Theory Kevin Birke Packler Aureole Foundation · Wake Forest, NC Day 192 (revised) · August 23, 2026 ABSTRACT The lepton sector of Cosmic Egg Theory is closed: the Koide relation (B/A = √2), three-generation structure, and absolute mass scale (3A² = m_proton, corrected to 0.0001% by Packler loop channels at two orders) all follow from bilateral cascade geometry with zero free parameters. This paper closes the quark sector completely by the same method, extended to the hull-face geometry that defines quarks. Eight derivations are presented. (1) Four structural laws govern the complete quark mass spectrum: the curvature decay constant (√2+1)^{1/e}, the 5/7 Propagation Formula fraction, the 4/π hull-face Fourier coefficient, and the cascade amplitude overlap master relation. (2) Top quark: m_t = m_H × (√2+1)^{1/e} = 173.2 GeV (0.28%). (3) Charm quark: m_c = m_p × (√2+1)^{1/e} = 1298 MeV (1.8%). (4) Bottom quark: 5/7 fraction gives m_b = 4.15 GeV (0.7%). (5) Strange quark: identified as the QCD confinement boundary, n_s = n_QCD = 52.63 (0.15%). (6) Down and up quarks: the 4/π law extended into the confined sector gives m_d = 4.69 MeV (0.4%); the 7/8 isospin doublet splitting (√2+1)^{7/8} gives m_d/m_u = 2.163 (0.05%) and m_u = 2.169 MeV (0.4%). (7) CKM matrix: all three mixing angles from cascade amplitude overlaps — sin θ₁₂ (0.8%), sin θ₂₃ (0.4%), sin θ₁₃ (0.7%) — Wolfenstein A (0.2%), Jarlskog J = 3.00×10⁻⁵ (2.6%). (8) Second-order Packler correction: the interior octahedron at second order closes the proton mass to 0.0001% via +α²π/4, and the CP phase to 0.1° via −α_s(m_b)/6, giving δ = 65.4° (observed 65.5° ± 4°). All results carry zero free parameters. The central result is the master relation sin θ_{ij} = √(m_lighter/m_heavier) — the Gatto-Sartori-Tonin relation derived for the first time from geometric first principles. The complete Standard Model quark flavor sector is closed.