An Equivariance Principle for the Axiomatic Method

A G-equivariant map R: X → T must send each x into Fix_T(Stab_G(x)); the set of all such maps bijects naturally with the product over orbit representatives of Fix_T(Stab_G(x)). An empty factor is impossibility, a singleton is characterization, a larger factor parametrizes a family. The bijection has been folklore since Eckmann (1954). We build on it a rigidity taxonomy of output spaces and an additive refinement theorem via the coinvariant Grothendieck group (X^gp)_G, exhibiting the impossibility-characterization dichotomy of the axiomatic corpus as one fixed-set product with three regimes. Five classical results (May's theorem, Nash bargaining, the Shapley value, the anonymity-neutrality-resoluteness impossibility, and the constraint theorem for label-symmetric classification) are derived uniformly, and the recipe is applied cold to a novel problem.