First, as I've mentioned you in another recent post - I do want to thank you for the courtesy of your posts in this thread.
fiat is -- at least when I'm GMing -- simply a kind of failsafe that I will only use if I feel that it would greatly improve the play experience or campaign. I'll give you an example - not a real example, but the type of situation where I will fiat.
I often resorted to some kind of fiat in 4E combat, when it got "grindy." Even when we become proficient at it, reduced monster HP, etc, there was invariably a point in which the outcome of the combat was 99.99999% certain: the monster was going to die. Let's say the monster has 33 HP left and a rogue does a rather dramatic sneak attack and does 29 HP of damage...I would often call that a kill shot.
Another example: There's about an hour before the end of a session, but there are several rooms before the PCs get to the room with the treasure and/or big bad guy. I might have one or two of those in-between rooms cease to exist, because I think the game experience would be better served by reaching the final room before the session ended.
I see important differences here that I don't think you do. I think that is probably connected to other ways that we talk about RPGing.
Deciding to give the players a win is a form of saying "yes". It's therefore not the sort of GM veto I am expressing a dislike for. Personally I don't do it in my 4e combat - if the monster has 4 hp left (or even better, 1 hp left) and therefore gets another turn, I like to taunt the players about that. That is because my own preference is to "say 'yes'" only when the stakes are not low, and as I experience the play of 4e, once the combat mechanics have been invoked, the stakes are not low.
If I were to say "yes" in this way, then I would tell the players. Part of what I see as distinguishing "say 'yes' or roll the dice" from illusionistic GMing is that the GM doesn't conceal his/her methods from the players.
The example of the rooms is to me quite different, as it does not involve action resolution at all. That is all GM framing. In a classic dungeon crawl game, rewriting the map like that would be a species of cheating. In my own games, given that they are not classic dungeon crawl games, there is no map in the relevant sense and I frame scenes as we go along. So what you describe is completely unremarkable to me.
Neither is an example of using secret backstory to block a declared action, which is what I was referring to in the post you quoted when I talked about
the GM being bound by action resolution. And said that deciding that the map isn't in the study
ndependelty of action resolution is no different from deciding that the NPC wins the race
independently of action resolution.
what about rolling random encounters - if I roll something that I think would be boring or tedious or detract from the game experience, I might roll again or choose something. Similarly with treasure.
Neither of these is an example of action resolution. They are both examples of content introduction.
Gygax talks about this in his DMG - I quoted the passages, and discussed them, in another recent thread (it is on the last page of the "What is XP worth?" thread). He expressly contrasts (though without using this terminology) rolls to introduce content (like wandering monsters, or finding a secret door that leads to an exciting part of the dungeon) with action resolution rolls that resolve a conflict (like fighting wandering monsters, or escaping fromm them, or being killed by a monster), and encourages GMs to manage the first set but not to interfere with the second set (except perhaps to mitigate the long term, but not the short term, consequences of a well-played PC being dropped to zero hp), as that would be "contrary to the major precepts of the game."
In most of my GMing I don't use random content introduction, as it seems to serve no useful purpose: in accordance with the "standard narrativistic model" I introduce content that will speak to the concerns/motivations/thematic drives of the PCs (as evinced by their build and play).
In my Traveller game I do use random content introduction, because that is a big part of Traveller. The most recent example was the roll to encounter a starship while leaving a system. I had one of the players make the roll. It turned up a pirate cruiser; I interpreted that in a way that made it fit into the ongoing action of the game (ie a vessel connected to the bioweapons conspiracy that that PCs are investigating and (perhaps) trying to thwart).
All this is orthogonal to my reasons for no liking "omnipotent" GMing, which is about how pre-authored content is used to constrain action declarations, not about how content for new scenes/situations is established.