So this is just something I was thinking after watching a guy playing LF so I could reacquaint myself with the series before NWoS comes out. Being my favourite Layton game, while I was watching through it, I managed to pick up something which eluded me in my initial playthrough of the game, perhaps buried underneath how sad I was feeling in the ending.
The game's narrative seems to take heavy inspiration from the five stages of grief.
Let's break this down.
Denial
This is the most obvious one. Hell, we already saw this through Layton's demeanour! It's mentioned throughout the early game that several NPCs are scared of his hat, already tipping us off that the status quo is trying to be challenged. Even in earlier games in the trilogy Layton mentions that he will refuse to take off his hat, even in situations where someone demands him to take it off, and that "a true gentleman never takes off his hat".
The truth is, Layton's gentlemanly behaviour is only really protecting Luke and that he's still upset over Claire's death. We learn in the game that the hat was a gift from Claire and that he's honouring her side of the bargain by refusing to take it off; in a way, it represents Layton's refusal to move on and repeatedly trying to deny the fact that Claire has died. In fact, the gentlemanly thing to do when going indoors is actually to remove the hat!
Even Layton being assaulted and silenced by the political authorities while they tried to cover it up is part of this. They're denying the truth that Layton is determined to uncover, and ultimately resulted in the actions of Dimitri and Clive, both of them putting up guises as "future Layton" and "future Luke" respectively to lure Layton into the fake London.
Anger
This is mainly manifested through Clive's actions. While Layton lost Claire in the time machine explosion 10 years prior, so did Clive lose his parents. Refusing to accept this, Clive's loss of his parents turned to anger and obsession over trying to find out who caused their death, to the point he became a reporter. In fact, this eventually resulted in him building the fake London, kidnapping a bunch of scientists to make them believe they were in the future, supposedly giving them motivation to work on what they believed was a time machine.
In fact, some of the scientists were actually working on a weapon of mass destruction with the aim of wiping out the real London. According to Clive, it was "justice - over single-minded scientists and corrupt politicians" and really shows how grief over losing a loved one can manifest into deep-seated revenge.
Bargaining
Now as for this, there's multiple ways this can play out. The most obvious instance of this is at the end of the game where Layton refuses to accept that Claire has to go back to her own time, however that's more depression. In fact, it's the aforementioned time machine project which is basically bargaining. Dimitri and Clive both lost loved ones in the time machine explosion, but instead of accepting the loss, they're trying to undo it.
Trying to mess with the laws of spacetime and several human lives just to undo one single tragedy is a recipe for disaster, and unfortunately neither party succeeds. This may not seem like a lot, but this mere act of bargaining is actually the main catalyst which kickstarts the game's plot, and there's multiple references to the time machine throughout the game.
Depression
The other obvious one. Notice how I said earlier that the game tries to challenge the status quo? Well, the weight of the final goodbye with Claire breaks Layton, challenging the whole thing with him being a gentleman. For a split second, we learn that this was actually just a mask to hide his emotional pain. At that instant, Layton is no longer a gentleman, he's now a broken man. This magnifies the sorrow of this scene, because we've never seen Layton like this before.
The feelings of loss also continue later on because Luke then leaves. This is foreshadowed during the cutscene where Luke gazes up at the statue, however it seems to mirror Layton's depression over losing Claire. Again, the gentleman mask slips for Luke, with him outwardly stating he's "not a gentleman yet", but for Layton this also means further loss; if you've played SC you'll find out he's now losing Clark Triton - one of his best friends.
In fact, an overarching theme in the narrative is that the main characters can't alter the past, and now have to live with the consequences of their actions; not only does Layton lose Claire, Clive is also arrested for his crimes.
Acceptance
At the very end of the game, Layton doesn't try to chase down Claire after she leaves, he just stands there and weeps openly. By this point, the mask has well and truly broken. Layton removes his hat out of respect, finally accepting that there is no way to bring Claire back. It's quite notable here because as mentioned earlier, Layton's hat is a symbol of his denial and refusal to move on from Claire. With the hat coming off, it's a sign of acceptance.
There's also a post-credits cutscene showing Layton in his office without his hat on, and he seems to have healed from his wounds. This further represents the fact that he's well and truly moved on from her, and had enough time to process his loss.
So yeah, that was quite a fair amount to unpack. The whole thing really shows how multi-layered LF's story is and I really appreciate it for challenging the status quo and showing the vulnerabilities of the main cast; the fact that it keeps them in London also helps with this because it allows more time to develop the characters, as opposed to also trying to balance it out with developing new locations like other Layton games have done.
TL;DR: Lost Future's overarching plotline is a rather deep and profound tragedy which illustrates grief through its characters, setting and overall narrative.
With that, I'll leave it with this poignant quote from Claire:
"I know you. And I know you'll stay strong. After all, that's what a gentleman does. I must go now, Hershel. Thank you for everything. Goodbye."
