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 $B^0 \to J/\psi K_S$ decay directly from the topological $\mathbb{Z}_3$ 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 $\mathcal{A}_{CP} = \mathrm{Im}(\omega) |V_{us}||V_{cb}||V_{ub}| \approx 2.926 \times 10^{-5}$. 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 $\mathbb{Z}_3$ root of unity ($\omega$). We prove that $\mathrm{Im}(\omega) = \sqrt{3}/2$ 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 ($A_\mathrm{seam}$) cannot be made unitary by any field redefinition ($\|A_\mathrm{seam}\|_F^2 \approx 1.035 \neq 3$). This non-unitarity is not a deficiency but a structural necessity; $A_\mathrm{seam}$ 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 ($|V_{ud}|$, $|V_{cs}|$, $|V_{tb}|$), 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 $|V_{cd}| = |V_{us}|$, matching experimental data to 0.16%. Furthermore, the framework uniquely determines the CP phase $\delta$ to be 117.2° ($\sin\delta = 0.8896$, within $2.1\sigma$ 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 $\mathcal{A}_{CP}$. Because the $\mathbb{Z}_3$ 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 $|V_{ub}|$ identified in Paper V, marking it as an irreducible gap at the current next-to-leading-order (NLO) expansion of the IGPS framework.