Again, we have similar sorts of reasons for doubting the existence and/or utility of the spectrum that
@Xetheral has posited.
I just want to focus on the "no notes, so wing it" aspect if the players say "We cross the hills to the north".
In this case, the GM ceases to exercise situational authority thus:
draw on pre-authored background, with latent situation, to frame the concrete scene.
Instead, the GM exercises situational authority thus:
draw on pre-authored background, and resulting sense of situation/verisimilitude, plus details of player action declaration, to frame the concrete scene.
These are quite different ways of resolving the action declaration "We cross the hills to the north". I don't see the analytic utility of putting them on the same spectrum. If the spectrum is measuring
likelihood the game will grind to a halt if the players declare "We cross the hills to the north" then I can see it might be measuring something, but I'm not sure that's a thing that it is useful to measure. It doesn't capture the difference between (eg) sandboxing and railroading, because a railroading GM can also just make stuff up when the players have their PCs cross the hills to the north, and tell them that story spontaneously. Lewis Pulsipher complained about this aspect of
playing D&D as a "living novel" over 40 years ago.