Paper2_Quark Mass Hierarchy and CKM Mixing in the β Framework

This paper develops the quark block of the beta framework with a deliberately narrow scope: the visible quark Yukawa hierarchy, the flavor-space mass matrices, and the emergence of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The starting point is a structural law where the channel beta organizes the Yukawa spectrum through an exponential ordering controlled by sector amplitudes and structural indices. The manuscript follows a step-by-step methodology: first, mapping observed quark masses into effective structural indices; second, explicitly writing the quark Yukawa Lagrangian and the resulting mass matrices (Mu and Md) after electroweak symmetry breaking; and third, deriving the CKM matrix from the mismatch between the up and down sector diagonalizations. A controlled benchmark branch is presented that reproduces the observed hierarchy of |Vus|, |Vcb|, and |Vub|, together with a nonzero quark CP phase and the corresponding Jarlskog invariant. The result is a disciplined effective realization showing that the quark mass hierarchy and CKM mixing can be organized inside one common structural language.