"It's been too long now"

At some point the stretch crosses a line and a new belief takes over from the meaning-to. It's been too long. A reply now would be strange. It would draw attention to the gap in a way that leaving it doesn't. Better to wait for a natural reason, a birthday or a piece of news, and start fresh from there. This belief is the one that turns a slow reply into no reply, so it's worth looking at closely.

Peggy Liu, SoYon Rim, Lauren Min and Kate Min ran a series of preregistered experiments on what people expect when they get in touch with someone, a short message to see how they're doing, or a small gift to show they'd been thinking of them. Across the studies, people underestimated how much the person on the other end appreciated it. The gap was robust. And two things made it larger. The appreciation was underestimated more when the contact was a surprise, and more when the two people were less close. The researchers' account is that the person receiving the message is focused on the surprise of it, and the surprise feeds the appreciation, while the person sending it is focused on something else.

Turn that towards the stretch. A reply after three weeks is more of a surprise than a reply after a day. A friend you've half lost touch with is more distant than one you spoke to last week. By this finding, those are exactly the conditions under which you most underestimate how glad they'd be to hear from you. The belief says the length of the stretch makes contact worse. The research says it makes contact more welcome, and makes you less able to see that.

So where does the strangeness come from? It's the surcharge, and the surcharge is yours. You're the one who feels the reply now needs to open with an explanation, carry an apology, and be long enough to make up for three weeks. Your friend isn't waiting for any of that. If they're waiting for anything, it's to hear from you. The apology and the explanation are things you'd add to make yourself feel better about the stretch, and the friend would read past them to the part that matters, which is that you wrote.

I'd put it this way, and it's an interpretation. There's no length of stretch after which a message is unwelcome. There's only a length after which you've talked yourself out of sending one. The second is real. The first isn't. And the reply that finally closes a long stretch doesn't need to be a good one. It needs to exist.