Because Satan Sneezes Too

Even when they weren't speaking, Alhaitham dogged Kaveh's every step through published responses to Kaveh's research articles in academic journals. He insisted on keeping a line of communication between himself and Kaveh open, even if the only way to do that was through very public ideological clashes. Pulling Kaveh's pigtails to get his attention lolol. It's implied that, for at least the few years between their fight and Kaveh moving in, this was the only communication between them--Alhaitham's refusal to allow their connection to entirely fade away. (And the fact that this is revealed in Kaveh's character stories--through his precious journal that records the moments of his life that had the most impact on him--shows just how deeply he values the fact that Alhaitham didn't give up.)

Another relevant side note: Alhaitham never asked Kaveh to give up his half of their house. Knowing half of it belonged to Kaveh, knowing that Kaveh may one day want to reclaim his part of it, knowing that it was listed as theirs, Alhaitham moved into the house and made zero effort to change its ownership. He was completely fine with living in "his and Kaveh's house." The stories suggest it was only months later (or even longer) that Kaveh even noticed he had the house, and he transferred away ownership of his portion without Alhaitham ever asking him (or even seemingly wanting him) to do so.

Please, let that sink in. Alhaitham actively left his grandmother's (presumably comfortable) house to move into "his and Kaveh's house," with no apparent explanation for why, and after doing so, he made no attempt to change that "his and Kaveh's" label. He moved into the house with no promise that Kaveh wouldn't show up on the doorstep the very next day and move in too. It almost feels like another deliberate provocation--I've moved into our house, are you going to come stop me? LBR, if Alhaitham had had his way, Kaveh would have been living there with him from Day 1...

There's also the fact that Kaveh literally can't write on a single message board anywhere in the entire nation of Sumeru without Alhaitham hunting his messages down and responding to them (which absolutely no one else does, by the way).

"NUH-UH!" "UH-HUH." "NUH-UH!"

Alhaitham's own character stories tell us explicitly that one of Alhaitham's defining character traits is "He is never where you need him to be," yet somehow...

Shot, and chaser:

Any time Kaveh is in the slightest bit of need or danger or just wants Alhaitham near, Alhaitham is "coincidentally" exactly where Kaveh needs him to be, whenever Kaveh needs him to be there.

Alhaitham didn't just "happen" to run into Kaveh in Port Ormos, an entirely different city from where he was supposed to be working. He didn't just "happen" to read the same terrible book as Kaveh when we know he otherwise would not waste a moment of his time on poorly-written literature...

He didn't just "happen" to appear when Kaveh was upset and needed a distraction in the House of Daena during Kaveh's hangout. He didn't just "happen" to be sitting around waiting when Kaveh needed answers after the Archon Quest. He didn't just "happen" to find Kaveh's academic publications and every single message board posting and respond to them at length and in public.

Which is exactly what Kaveh's mother told Kaveh he needed.

What level of down bad is "Abusing your powers as an Akademiya employee to keep tabs on your crush's library loans"? Just asking for a friend.

The only person for whom Alhaitham just "happens" to be available is Kaveh, over and over and over again--because he is very deliberately making himself a constant presence in Kaveh's life.

(Like, out of all things, I think people really underestimate the devs deliberately paralleling the romantic relationship between Kaveh's mother and father with Kaveh and Alhaitham's relationship. If you want to point to one thing that says "These two characters are intentionally queer-coded," it doesn't get any more obvious than this.)