One of the main goals of Flavor Physics is to overconstrain the CKM matrix. Non-perturbative QCD, however, makes this challenging by introducing hadronic matrix elements that cannot be reliably calculated. In our work, we use the approximate U-spin symmetry (unitary rotations between d and s quarks) to get around this problem. Our study reveals a rich mathematical structure underlying the sum rules that allows us to formulate an algorithm for deriving all U-spin amplitude sum rules to any order of the symmetry breaking. This novel approach to deriving the sum rules does not require one to perform any explicit calculations, and allows for simple diagrammatic interpretation. In my talk, I will describe these results and argue that, despite the relatively large breaking of the order 30%, the systematic expansion in U-spin breaking can be used for precision physics. (The talk is based on arXiv:2205.12975.)