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The basic game text goes out of its way to convince us that the house is Alhaitham's--it is literally called "Alhaitham's Home" even during Kaveh's hangout.

But in-universe, we never see Alhaitham in his house unless Kaveh is there.

Throughout Sumeru's archon quests, we run into Alhaitham practically everywhere: in Port Ormos, in Aaru Village, in the Akademiya, at the Grand Bazaar. But never, despite the fact that the story wants to sell him as a homebody, in his own home. In fact, the only references we have to Alhaitham's house at all throughout the entire lead up of the archon quests are mentions of Kaveh.

Alhaitham doesn't suggest going there, doesn't ever go home himself (at least that we're explicitly told about) until the end of the archon quests, and has no cutscenes in his own house until Kaveh returns from the desert.

Then, all the sudden, Alhaitham literally can't wait to go home and walks off in the middle of a conversation.

From that point on, every time we're permitted to enter Alhaitham's house, it's either when he and Kaveh are together or with Kaveh alone.

Even during Alhaitham's birthday skit of all things, they went out of their way to tell us Kaveh was there:

The map tag tells us the building is Alhaitham's Home, but the story reinforces a different idea:

Goodbye ,

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