CP Phases, Topological Holonomy, and Non-Unitary Geometric Flavor Mixing

Description: This paper, the sixth in the Information-Geometric Physics System (IGPS) series, derives the CP-violating amplitude for the B⁰ J/ KS decay directly from the topological Z₃ seam holonomy. Building upon the spectral geometry established in Paper V, this work demonstrates how complex phases and physical unitarity emerge from geometric couplings with zero free parameters. Key Findings: CP-Violating Process Amplitude: The framework predicts the time-integrated CP asymmetry A₂₏ = Im () |Vₔₒ||V₂₁||Vₔ₁| 2. 926 10^-5. This parameter-free derivation agrees with the PDG world average to within 5% with the correct sign. Topological Exactness of the Phase: The CP phase originates from the primitive Z₃ root of unity (). We prove that Im () = 3/2 is a topological invariant that survives Kaluza-Klein reduction to four dimensions and is exact to all perturbative orders. The Frobenius Norm Obstruction: We rigorously prove that the bare seam amplitude matrix (Aₛeam) cannot be made unitary by any field redefinition (\|Aₛeam\|F² 1. 035 3). This non-unitarity is not a deficiency but a structural necessity; Aₛeam functions as a localized geometric Yukawa-like coupling rather than a physical rotation matrix. Restoring Unitarity via Horn's Theorem: By applying probability conservation to the seam crossing amplitudes, we derive the diagonal CKM elements (|Vₔ₃|, |V₂ₒ|, |Vₓ₁|), matching PDG values to better than 0. 01%. The resulting magnitude matrix is doubly stochastic, and Horn's theorem mathematically guarantees the existence of a fully unitary CKM matrix. Parameter-Free Predictions: The distance-matrix symmetry yields the parameter-free identity |V₂₃| = |Vₔₒ|, matching experimental data to 0. 16%. Furthermore, the framework uniquely determines the CP phase to be 117. 2° (= 0. 8896, within 2. 1 of the PDG value) solely from the geometry and topology of the defect. Residual Gaps and Outlook: The paper candidly addresses a 5% residual gap in A₂₏. Because the Z₃ phase is topologically protected, this gap cannot be closed by perturbative corrections to the holonomy phase. It is traced back to the 2. 72% geometric discrepancy in |Vₔ₁| identified in Paper V, marking it as an irreducible gap at the current next-to-leading-order (NLO) expansion of the IGPS framework.