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

We define a fixed-input phenomenological ordered-rotation ansatz matrix for the Cabibbo–Kobayashi–Maskawa (CKM) sector. The construction is motivated by the down-type Dual Koide benchmark and by a Pontecorvo–Maki–Nakagawa–Sakata (PMNS)-motivated split-sector ordering template, but neither ingredient is treated here as a first-principles source of quark mixing. The ansatz considered here is specified by the M1 down-type benchmark, the sine-element half 1–2 split rule, the sine-half 2–3 split rule, and an empirically imported up-sector 3–1 square-root-coordinate correction $R_{u31}^{(\sqrt{m})}$ with $\delta_{u31}=\tfrac{1}{4}\pi$, 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 enters as the Hermitian-conjugated reversed 3–1 terminal step rather than as part of a single global cyclic ordering. The superscript $(\sqrt{m})$ 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 $$|V_{\mathrm{CKM}}^{\mathrm{ans}}|\simeq\begin{pmatrix}0.974455 & 0.224552 & 0.003754 \\0.224408 & 0.973605 & 0.041646 \\0.008880 & 0.040861 & 0.999125\end{pmatrix},$$ with Jarlskog invariant $|J|\simeq 3.24\times 10^{-5}$, $(\alpha,\beta,\gamma)\simeq(85.81^\circ,22.97^\circ,71.22^\circ)$, and $\sin(2\beta)\simeq0.719$. The result is a compact phenomenological target for future flavor-model construction, not a microscopic flavor theory and not a CKM-likelihood fit.