In this thesis, I present my work on higher-order flavor amplitude sum rules. I develop a systematic study and a unified framework for deriving SU(2) flavor amplitude sum rules, with a focus on U-spin, highlighting a rich underlying mathematical structure that governs these relations. This approach circumvents the need for explicit calculations and admits a clear diagrammatic interpretation. Several general results are proved, and the method is illustrated with experimentally relevant examples, including higher-order amplitude sum rules for multibody hadron decays and CP-violating charm decays.