What is the Arcana check for? I don't see an action declaration from the wizard in your breakdown.
I'll assume that was a serious question. For seeing if the wizard's character does actually know that knowledge. Arcana is the skill linked with knowledge about elementals and their strengths and weaknesses after all. And as a DM, I can call for checks, correct?
That's not the DM's problem. It's up to the players to play their characters effectively.
I'm not saying it is a problem, but you keep using it as a defense. Everything is fine, because the smart play is to verify. But, just because it is smart does not mean that is what the player will do.
And you know what is a DM problem? The players not having fun. Which is something which I could see happening in extreme cases of this whole discussion.
My players do because they have an incentive to. As an example from my current Eberron campaign, the players found a chamber in the dungeon containing crates covered in brown mold. I telegraphed an unusual chill in the adjoining chamber. A couple of characters ran afoul of it and took a bit of cold damage when they kicked down the door to said chamber. Everyone in my group is an experienced player. They knew this was brown mold and how to do deal with it (cold damage) and how not to use fire on it. But, they know that I change things up from time to time and, with the wizard having no cold-damage cantrips and only one spell slot remaining, they could not take any risks on this.
So the wizard used mage hand to collect a small sample of the brown mold, not enough to do damage to anyone, with a test tube. He handed it off to the warforged fighter who has integrated alchemist's supplies. Ten minutes of testing and analysis, a wandering monster check (no wanderer), and a successful Intelligence (Alchemist's Supplies) check later, they verified it was brown mold. The wizard cast an ice knife spell, destroyed the brown mold, and they were able to obtain the schema they were seeking to complete their quest.
The players chose to play effectively. All I had to do was describe the environment and narrate the results of the adventurers' actions.
What you described isn't incentive. They were cautious, yes, but if later in the campaign they encounter a brownish mold in a cold room will they check it again? How about the third time? The Fourth?
Sure, you might change it, and then you'll telegraph it by describing something different about the mold this time, maybe by adding yellow stripes to it or something.
But, despite their caution, look at how you describe the thought process. Cold room, brown mold. It is probably brown mold,
but we don't have more than one spell slot and no cold-based cantrips so we should verify it is brown mold first. The entire verification was based upon their lack of resources, if they had had a cold cantrip, they would have just shot it at the mold, because there would have been no loss of resources to the party. Unless you had changed it to do something else. That is what I've been talking about. Verification will not happen every time.
I can't speak for any one else, but for my part its because I repeat the same things over and over and they just bounce off. I have a hard time believing that you aren't at this point able to answer your own questions. I mean just considering what you've now posted, the answers to your own questions are present if you are willing to see them. I admit I have weird pet peeves and my social-emotional framework doesn't well align with the rest of the human race, but honestly if you made attacks and cast open aspirations or said "You make me so angry", it would be less frustrating to me and more understandable than what you are doing.
I try not to assume what people will say, I can sometimes predict what an answer will be before I see it, but then I'd be talking to myself and not the actual other person.
I apologize if it frustrates you, but repeating the same thing over and over does not necessarily convince me of anything, and in fact, if I bring up a counter-point that doesn't get addressed, then it is nothing more than circular movement.
How is it that when you've well understood that people were saying "players have absolute authority over their character's thoughts and actions" that you've now added to that something of your own invention in order to condemn their position as illogical, namely that the players also have absolute authority over the character's background, and by which you mean something that they never said, that they also have absolute authority to create any background that they like at any time in the game?
I've added nothing, just followed the logic.
Absolute Authority over thoughts and actions is translating to absolute authority over the character's mind. That's what thoughts and actions are, since the Player does not have the authority to automatically succeed. This would then include authority over your own memories. Barring an outside influence, if I have absolute authority over my character's thoughts, then I should assume I have absolute authority over what they do and do not remember. We even include emotions in this, relationships.
That means they have the same authority over their past that they have over their actions, because they are the ones telling that story.
Now, you can lock backgrounds, tell the player that they are not allowed to alter or add to their background after the first session, but most tables do not do this. It is perfectly acceptable at a lot of tables to allow players to flesh out backgrounds over the course of play, because writing the full life story of a 25 year old soldier who lost his way in war and converted to the worship of a peace goddess... well that is hard.
Also, I just confirmed this character was 1) in a military unit, 2) fought in battles if not a full war 3) joined a temple or religion 4) worships a goddess of peace. All of that is setting information, the existence of these things should be in the realm of the DM, but, I have full control over my character's thoughts. I can say that my character believes in peace because he has buried too many friends, by the way, he know has dead friends.
I'm not being a problem player, I'm not going beyond the pale, but I've been adding to the setting. Now, the DM is fully within their rights to veto any of this, but if they, for example, say that there has been no war for the last 20 years, then my character drastically changes, because he can no longer have the thoughts he had about war, because he no longer has the experience of war. I did not have absolute authority over my character's thoughts, because the DM declaring a peaceful era immediately changed those thoughts.
So why is it surprising that someone who you admit said "players have "absolute authority" over their characters thoughts and actions" should think that absolute authority over their background is a step too far? And further, in the Francis example, we have gone even one step further past claiming that the player has absolute authority over their background, and are now asserting that the background has absolute authority over the setting.
Why should it even be confusing that someone who only started from the proposition "players have absolute authority over their characters thoughts and actions", should not able to answer your question regarding whether Francis exists in the city? After all, even if someone did assert that players had absolute authority over their background, that would only mean that the player could assert that Francis existed sometime in the city in the past. You could not assert on the basis of your authority over background, that now in the present Francis is still alive, still in the city, and still serving in the guard. All of those things could have changed between the point you asserted Francis had existed and the present moment in game, and regardless of your absolute authority over background you could not decide those things without absolute authority over the setting. So of course people can't answer your question in any general way or give you any other answer but "Maybe."
And remember, these people by your own admission never began by asserting players had absolute authority over their background in the first place.
In point of fact, while I've asserted that players do have a sort of absolute authority over their background, I asserted that only in the sense that a player character's background is inviolable. That is to say, a player may absolutely refuse any other participant's suggestion to alter their background. A GM cannot force a player to have a backstory they don't want. A player can say, "Mess with me. I want to have complications and drama because that's the sort of game I want to play.", and thereby give the GM permission to introduce backstory elements. But a player can also say, "My backstory is meant only to serve as backstory, and I only want my character to evolve through forestory, and not by making unwanted revelations about his past." All that is fine, but it is also very different from the assertion that a player has an absolute right to introduce backstory, much less that having introduced backstory, he has some absolute right to insist that present situations conform to his desires and expectations. Even if the player's relationship to Francis is inviolable and even if their is a table agreement to be "hands off" with respect to Francis, such a social contract does not mean Francis is here now in the present. The GM, being absolutely in charge of the setting, could say, "This guard isn't Francis. The Guard says, "So you're a friend of Francis? Yeah, he has the night watch tonight. I'm Robert. We agreed to switch because I'm going to see a lovely little lady tonight at the festival.", or any number of other things. Francis is after all, an NPC, whether he's in your backstory or not.
So, I think I cover some of this up above, but a few salient points.
There is no "sort of absolute authority". If authority is not absolute, then it is not absolute authority.
Secondly, while you very much could respond to the player searching the city for Francis with "Well, Francis is dead" that feels a bit... squicky, to me at least. "Hey, I want to look for an old friend in this city." "Okay, he's dead, let's move on"
Now, you can turn any of these into interesting points. Maybe with him moving you'll encounter him later because the player is curious why Francis decided to leave. Maybe with him no longer being a guard you can explore some aspect of the city or have a personal character building moment. Maybe with him being dead you can get a personal sub-quest to avenge your friends death. All of these can be interesting. All of them also mean the player changed the setting, because Francis did not exist until the player said so. The DM okayed it, the DM allowed it, but the Player changed the setting here, which is why I've said these lines are not burnt into the ground. Authority over thoughts leads to emotions and memories which leads to relationships with NPCs which leads to the player changing the setting.
In point of fact, the GM could say that. The GM could for example overrule a character whose IC motivation is to kill the other members of the party, or could overrule a character whose concept is that he's working for the bad guys. I'm not saying a GM should always do that, but it takes an extraordinarily mature group to deal with that in a cooperative fashion.
So, reading this it seems we are actually in agreement.
A DM can tell a player what their character does not do or does not think. Therefore their is no absolute authority. If you claim an absolute authority, then you must accept everything that flows from that.
I especially agree that a DM should be very receptive to players who point out aspects of their character or backstory that are inviolable. Making sure the players are invested and feel like their investment matters is key to a successful game.
It seems after reading this entire post that you think I'm being unfair by saying if you give a player absolute authority over their character's thoughts, that will apply to memories, emotions, and relationships as well. But, our thoughts are shaped by are past, by how we are raised, by what we feel. You cannot separate them.
One wonders if the person making these claims believes their own person extends to encompass all that they can observe or think on?
This made me chuckle, because I thought about someone who would claim to have absolute authority over their own thoughts. I find that idea to be wrong, we do not have that sort of authority over our own minds.