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@DanRamras That's my thinking exactly. When I was trying to understand the action on the set of orientations, I wrote it down wrong at first. Working out my mistake is what led me to notice that it seems simpler when we restrict attention to total orders.
Yes, I have noted that. I even dared to say that it is a simplification of that argument since I find it easier to think about the set of total orders as opposed to the set of all orientations.
