I'm confused why I've received pushback, then. If the GM's call is the only one that matters, this definitely changes the nature of the invisible rulebook and why it's invisible. Now, the only invisible rulebook that matters is the GM's, and since it's invisible this is hidden from the players. Yet you say that the GM must be able to explain their reasoning, which cuts against the invisible rulebook because it makes it visible. We're back to the competing ideas of transparent play vs hidden play and why you'd even use an invisible rulebook at all.
I don't think 'invisible rulebook' is a helpful concept for you in understanding FK. The person who came up with it, as a metaphor, did not mean anything like rulebook as you understand the term. From reading her essay, she meant a sort of headcanon guide book to genre/setting norms. I think it would be best if you forgot about 'invisible rulebooks' completely.
Re arbitrariness, if I as GM am operating within the internal aspect of an NPC, their actions do not feel arbitrary at all. Their actions result from their motivations, hopes, fears, grudges, loyalties etc etc. I know some people don't/can't get their head around this, cannot take on the internal aspect of another character (possibly not even their own PC), and don't think other people can, either. In which case they should not GM a character based FK/free-roleplay game IMO, and if they don't trust the process they should not play in one, either.
Edit: Another way of putting it: when GMing FK style, I as GM am normally in 'actor* stance' playing NPCs, and in 'neutral referee' or 'world stance' when adjudicating. I am never in 'author** stance', except in a very residual sense that when GMing in a particular setting/genre my refereeing necessarily takes account of the original setting authors' - the rules they made for the world.
*In the Ron Edwards sense of internal-aspect method acting.
**In the Ron Edwards sense of story-creation, what would make a good story.
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