The CP violation discovered in 1964 could not be explained with only the four quarks known then. Kobayashi and Maskawa expressed weak-interaction quark mixing with a 3×3 matrix—the modern CKM matrix—and showed that a complex phase in the matrix naturally produces CP violation. Introducing such a phase mandates six quarks, i.e., three generations. Within years, the charm, bottom, and top quarks were successively discovered, vindicating the theory. Today, precision studies of B- and K-meson decays test the CKM matrix and link to research on the cosmic matter–antimatter asymmetry.

In the Kobayashi–Maskawa scheme, weak eigenstates |d′,s′,b′⟩ relate to mass eigenstates |d,s,b⟩ via |d′⟩=V_ud|d⟩+V_us|s⟩+V_ub|b⟩, with V the CKM matrix. V is an SU(3) matrix characterized by one physical complex phase δ and three mixing angles. A non-zero δ yields CP violation, observed in B^0–^0 mixing and K_L→ππ decays. Unitarity-triangle analyses measure |V_{ub}|, sin2β, etc., and the BaBar/Belle experiments confirmed the theory. The CKM matrix exhibits a hierarchical structure expressed by the Wolfenstein expansion λ≈0.22. The CP violation magnitude is insufficient for baryogenesis, motivating searches for physics beyond the Standard Model.

The Jarlskog invariant J=Im(V_{us}V_{cb}V_{ub}^*V_{cs}^*)≈3×10^{−5} quantifies CP violation and is non-zero only with three or more generations—a key KM insight. B-factory data on Δm_d, Δm_s, and A_{CP}(B→J/ψK_S) close the unitarity triangle in the (ρ,η) plane. Global fits including NNLO QCD corrections confirm consistency among |V_{ub}|/|V_{cb}|, γ, and β. SUSY or Z′ extensions could appear as deviations in FCNC processes or additional CP phases; Belle II and upgraded LHCb aim for higher sensitivity. For electroweak baryogenesis the CKM phase yields η_B~10^{−20}, far below observations, necessitating new CP sources.