Phenomenological Ordered-Rotation Ansatz for the CKM Matrix

We define a fixed-input phenomenological ordered-rotation ansatz matrix for the Cabibbo–Kobayashi–Maskawa sector. The construction is motivated by the down-type Dual Koide benchmark and by a PMNS-motivated split-sector ordering template, but neither ingredient is treated here as a first-principles source of quark mixing. The single object tested below is the ansatz matrix specified by the M1 down-type benchmark, the half-Cabibbo 1–2 split rule, the sine-half 2–3 split rule, and an empirical imported up-sector 3–1 square-root-coordinate correction Rᵤ₃₁(√m) with δᵤ₃₁=1/4π, interpreted as a split-sector 3–1 phase-sharing input. The product is mirror-like in the 1–2 and 2–3 particle/flavor blocks, while the imported square-root correction is attached as the reversed 3–1 terminal step rather than as part of a single global cyclic ordering. The superscript (√m) is part of the notation and indicates an auxiliary square-root mass-coordinate correction, not an ordinary particle/flavor rotation. 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 |J| 3.24× 10⁻⁵, (α,β,γ)(85.81∘,22.97∘,71.22∘), and sin(2β)0.719. The result is a compact phenomenological target for future flavor-model building, not a microscopic flavor theory and not a CKM-likelihood fit.