Move a dollar from a richer person to a poorer one without reversing their order, and EVERY honest inequality measure must fall — at once. It is the one axiom they all obey. Slide (or tap) the transfer and watch three different indices drop in lock-step.
The Pigou-Dalton transfer principle, shown on three unrelated indices at once. A rank-preserving, mean-preserving transfer from a richer to a poorer person reduces inequality — and any measure worth the name must register the drop. The slider performs exactly such transfers (levelling toward the mean) and computes the Gini, the coefficient of variation (σ/μ), and the Atkinson index (ε=1) live; all three fall together and reach zero at equality. A fail-loud self-check throws unless every one of the three is strictly lower at a partial transfer than at the start.
The transfer here is progressive levelling toward the mean (a composition of Pigou-Dalton transfers), and the three indices are a representative sample of many; the mean is held fixed. Each index’s arithmetic, and the co-monotone drop, are exact.