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we immediately get a ton of details on luke's current situation. his mother has been away for quite some time and he almost completely refuses to talk to his father. layton has only ever seen luke as a baby, so neither would know each other personally at this point; in any event it is concluded that he might be willing to connect with someone outside his family (which suggests familial conflict).
and luke, just like how we've seen him in the main series, is usually bright-eyed and bushy-tailed: he's only so withdrawn now. the maid suspects this is because of the absence of his mother, which is interesting, as this being just a suspicion suggests that luke himself has said nothing explicitly to that end - even if it is true, luke has not communicated this. and indeed, he acts very grown-up: it is unclear if this is a behavior he's always exhibited, or one only picked up in recent events (relating to his mother, the specter, or something else).
despite neither layton nor luke knowing each other personally, luke seems to know a great deal about layton - calling him "esteemed", and knowing his aptitude for puzzles. as it is, luke was the one writing for layton's assistance, so he clearly knows much about layton's ability to solve baffling mysteries. and isn't it fascinating how their relationship began over luke seeing layton as this genius celebrity? that surely leads into the way luke would proceed to idolize layton, and call himself "layton's apprentice" (over "layton's friend").
clark, upon learning that luke wrote the letter in his name, immediately lashes out at luke. he seems to care about luke, sure, but he's incredibly dismissive and easily assumes the worst. clark doesn't give luke the benefit of the doubt, or even the time of day to explain himself - so no wonder luke has refused to talk to his father! he wouldn't be allowed to get a word in edgewise anyway! (i apologize for immediately seeing the worst in clark, but immediately yelling at your son for doing something kind of bad before even listening to him is a red flag to me...)
and even after layton gets him to settle his anger, clark continues to be dismissive and reductive. of course, he has reason to question the prophecies his son claims to give, but it doesn't even seem like he's tried to ask luke about them: he just calls the whole matter ridiculous and leaves it at that. but then, interestingly, clark blames himself for this behavior... firstly, this is further evidence of his dismissiveness towards luke, as he sees his supposed soothsaying as something to be blamed for; secondly, in some way we do not yet know (he changed the subject and left before elaborating), he feels responsible for luke's current state, as if he's aware of his shortcomings as a father.
this fellow seems to be friends with luke's mother, and finds it odd that she's left for so long without speaking a word, suggesting that perhaps she's had a spat with clark. notably, even though he proceeds to dismiss this, saying the two are a happy couple, luke briefly interjects as if his parents have indeed argued recently. in general, i'm inclined to believe luke's home life isn't the rosiest...