A Geometric CPVSM Framework: Dynamical Quark Mass Spurt with Explicit CP Violation in the Standard Mode

We present a top-down geometric framework within the Standard Model (SM), utilizing dimensionless 2D ratio vectors $(x,y)$ and $(x',y')$ to establish a Cartesian-like parameterization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This approach provides an intuitive physical mapping that explicitly disentangles up- and down-type quark sector contributions to electroweak mixing and CP violation. Crucially, we prove an exact algebraic dimensional reduction from the 9 real degrees of freedom of the complex mass-squared matrix $\mathbf{M}^2$ down to 5 baseline parameters, and ultimately to 2 invariant ratio coordinates ($x,y$) in the unitary diagonalization matrix $U$. This baseline yields exact analytical expressions for the CKM matrix and a closed-form Jarlskog invariant $J$, achieving robust numerical convergence to $|J_0| \approx 3.08 \times 10^{-5}$. We demonstrate that the exact commutativity condition $[\mathbf{M}^2_R, \mathbf{M}^2_I] = 0$ is the fundamental origin of an intrinsic fourfold magnitude degeneracy governed by $|p| = |p'| = |p^*| = |p'^*|$, inducing non-perturbative Wolfenstein scale mixing that signals the necessity of non-commuting quantum corrections ($[\mathbf{M}^2_R, \mathbf{M}^2_I] \neq 0$). Remarkably, near critical boundary geometries ($xy \to 0$), the system exhibits a \textit{quark mass spurt} that reshapes the mass hierarchy while geometrically preserving non-collinear CP violation ($xy' - x'y \neq 0$), yielding magnitudes sufficient to account for the cosmological Baryon Asymmetry of the Universe (BAU). Extending this vector mismatch paradigm to the PMNS lepton matrix with Dirac neutrinos ($\nu_R$), we naturally accommodate large leptonic mixing angles through angular vector offsets, establishing a unified, rephasing-invariant geometric framework for flavor origins across both quark and lepton sectors.