Quark Flavor Mixing from the Simplex Geometry of the Counted Patterns

In IDWT every particle is a pattern of identical portions, bound in d spatial directions and arranged on n levels. The number of portions is the count C = S(n, d) = C(n + d − 1, d), and the particle's mass is a family unit times that count. The count is the number of lattice points of a d-dimensional simplex of side n − 1. This paper shows that the quark mixing matrix follows from that shape. The lighter quark of each pair sits in a corner of the heavier quark of the same depth. The three mixing angles are the three corner overlaps, the square roots of the counts' ratios. The CP phase is the angle at which two sides of a regular tetrahedron meet. With five counts fixed by the masses and no fitted constant, the construction gives eight of the nine matrix elements within 2 percent, the Wolfenstein constants λ and A within 0.7 and 0.4 percent, and the unitarity-triangle angles α and γ and the CP strength J at the measured size. The same geometry gives the dominant pairing of the quark generations, and it places the bottom quark exactly inside a side of the charm pattern.