CKM substructure from the weak to the Planck scale

We look for relations among CKM matrix elements that are not consequences of the Wolfenstein parametrization. In particular, we search for products of CKM elements raised to integer powers that approximately equal $1$. We study the running of the CKM matrix elements and resolve an apparent discrepancy in the literature. To a good approximation only $A$ runs, among the Wolfenstein parameters. Using the Standard Model renormalization group we look for CKM relations at energy scales ranging from the electroweak scale to the Planck scale, and we find 19 such relations. These relations could point to structure in the UV, or be numerical accidents. For example, we find that $| V_{td} V_{us} | = | V_{cb}^2 |$, within $2\%$ accuracy, in the $10^9$-$10^{15}$ GeV range. We discuss the implications of this CKM relation for a Yukawa texture in the UV.