Phenomenological Ordered-Rotation Ansatz for the Cabibbo–Kobayashi–Maskawa Matrix

We define a fixed-input phenomenological ordered-rotation ansatz for the Cabibbo–Kobayashi–Maskawa (CKM) matrix. The construction is motivated by the down-type Dual Koide benchmark and by a Pontecorvo–Maki–Nakagawa–Sakata (PMNS)-motivated split-sector ordering template. The ansatz studied here is specified by the M1 down-type phase benchmark, a half-seed sine rule in the 1–2 sector with a tangent wrapper, a half-seed sine rule in the 2–3 sector with a sine wrapper, and an up-sector 3–1 square-root-coordinate correction Rᵤ₃₁(√m) whose magnitude |θᵤ₃₁| is fixed by mᵤ and mₜ, and whose phase is prescribed by split-sector phase sharing, δᵤ₃₁=1/2δν31=1/4π. The ordering is best viewed as a reduced six-step split-sector template: a formal down-type 3–1 → 1–2 → 2–3 sequence is recombined with the Hermitian-conjugated up-type sequence, while the formal down-type 3–1 square-root block is set to the identity and drops out of the ansatz product. The logical status of the inputs is classified explicitly in the paper. With the M1 primary input the ansatz magnitudes are|VCKMᵃⁿˢ|pmatrix0.974455 & 0.224552 & 0.003754 \\0.224408 & 0.973605 & 0.041646 \\0.008880 & 0.040861 & 0.999125pmatrix,with Jarlskog invariant |J| 3.24× 10⁻⁵, (α,β,γ)(85.81^∘,22.97^∘,71.22^∘), and sin(2β)0.719. Relative to the adopted comparison scales, all |Vᵢⱼ| entries lie within about 1.6 comparison-scale units, and the unitarity-triangle angles are broadly consistent with the observed ranges, with γ as the main residual tension. The result is a compact fixed-input phenomenological target for future flavor-model construction.