A Geometric Selection Rule for CKM Mixing from a Minimal Complex Hermitian Hamiltonian

Overview This paper presents a novel organizing principle for quark flavor mixing: CKM is selected by discrete geometry rather than fitted by continuous parameters. We introduce a minimal complex Hermitian Hamiltonian template with two sector-independent geometric anchors: ti/tr = 180/π (the canonical map between continuous and discrete angular measures) vbase = √15 ti (related to the 15 Weyl fermions per generation in the Standard Model) Key Results A full profile scan over 17 candidate ratio angles identifies a sharp minimum at θ = 30°, corresponding to: κ(d)₂₃ / κ(u)₂₃ = cos(30°) = √3/2 The global best fit achieves χ² = 0.287 against experimental data, reproducing all five rephasing-invariant observables (|Vus|, |Vub|, |Vcb|, |Vtd|, J) within uncertainties. Methodology Discrete lattice approach: 2,992 hypothesis points tested (17 angles × 16 rational depth values × 11 discrete CP phases) Minimal continuous fitting: Only two parameters (ε₁₂, κ₂₃⁽ᵘ⁾) are optimized per lattice point Robustness validation: Multi-start optimization, full-lattice toy experiments, and leave-one-out extrapolation Significance The appearance of √3/2 — the height-to-side ratio of an equilateral triangle — suggests that quark mixing may be constrained by fundamental geometric structures rather than arbitrary Yukawa couplings. This framework provides falsifiable predictions for future flavor measurements and motivates extension to PMNS (neutrino) mixing. Resources Analysis Code: Google Colab Notebook Related Work: Geometric Hamiltonian for charged-lepton mass ratios (Zenodo) Keywords CKM matrix, quark mixing, discrete geometry, flavor physics, geometric selection rule, Hermitian Hamiltonian, CP violation