Same-diff? Conceptual similarities between gauge transformations and diffeomorphisms. Part III: Representational conventions and relationism

The following questions are germane to our understanding of gauge-(in)variant quantities and physical possibility: in which ways are gauge transformations and spacetime diffeomorphisms similar, and in which are they different? Sophistication is the most popular attitude towards some of these questions: roughly, it takes models related by these symmetries to represent the same physical possibility. In the previous paper in this series, I discussed obstacles to sophistication and then showed how these obstacles are overcome by theories that fulfill three Desiderata (i-iii). But this resolution still leaves open two main worries about sophistication: (a) it allows the individuation of structure-tokens to remain intractably prolix and thus of limited use, which is why practising physicists frequently invoke 'relational, symmetry-invariant observables'; and (b) it leaves us with no formal framework for expressing counterfactual statements about the world. Here I will show that a third Desideratum, (iii), answers these worries. The new Desideratum requires a `relational' understanding \emph{of coordinates} (or frames, etc).