Finite support in a permutation action ($G_{(S)}\subseteq \mathrm{Stab}(x),|S|<\infty$)

Suppose a permutation group $G$ acts on an underlying atom set $A$ and on a collection of objects built over it. A finite subset $S\subseteq A$ supports an object $x$ when every permutation fixing all members of $S$ fixes $x$. Pairing supported objects uses the union of their supports. A definable subset or functional image inherits a support from its domain and parameters. In a hereditarily finite-supported Quine-atom model, every member is recursively required to have such a support; symmetry then rules out a choice function on the pairs of atoms.