I usually think of gauge freedom as saying “there is a family of configurations that all produce the same observables”. I don’t think that gives a way to say some configurations are correct/incorrect. Rather some pairs of configurations are equivalent and some aren’t.
That said, I do think you can probably do something like the approach described to assign a label to each equivalence class of configurations and do your evolution in that label space, which avoids having to pick a gauge.
How does this correctness check work?
I usually think of gauge freedom as saying “there is a family of configurations that all produce the same observables”. I don’t think that gives a way to say some configurations are correct/incorrect. Rather some pairs of configurations are equivalent and some aren’t.
That said, I do think you can probably do something like the approach described to assign a label to each equivalence class of configurations and do your evolution in that label space, which avoids having to pick a gauge.