Sets of cardinality seven are not sum-dominant

We give a self-contained elementary proof of the known result that every set $A\subset \mathbb{R}$ of cardinality seven satisfies $\mathrm{\mid }A+A\mathrm{\mid }\le \mathrm{\mid }A-A\mathrm{\mid }$. The argument adapts Hegarty’s method, using representation counts and the largest positive differences. An exact counting identity, two applications of the Cauchy–Schwarz inequality, and an analysis of a symmetric six-element set with one point added complete the proof, with no computer enumeration. This answers in the affirmative a question of Chu for the seven-element case.