An unintuitive endgame pattern

I was caught off guard in a recent game, and was a bit lucky to hold on to my advantage. My error was due to misunderstanding a seemingly trivial structure. The thumbnail's solution is at the end, but I think the journey will better explain why it matters and includes another nice little pattern. Endgames are best understood, not memorized.

I traded down into this endgame, knowing I’d lose a pawn but under the assumption that the conversion should be simple. Our kings get stuck competing for the same square, and with him to move I can just wait it out. Our pawn structures are identical. So surely if I just copy his moves we'd get stuck and he'd have to retreat?

Having reached this position, I realized I was mistaken. If I just continued with symmetry, White could force enough chaos (f3 then g4) that I'd have to let them close the structure, forcing me to move my king first. However, I happened to have a nice plan B

I get to steal zugzwang and the conversion is simple.

But what if I didn’t have that option? Well, that’s our thumbnail position. Both kings are genuinely stuck. When the kingside locks up, it's zugzwang for both players

g6 fails to the in-game plan.

f5 does block the initial plan, but by moving the pawn two squares you waste a tempo. And that's going to cost you the game

f6 (despite Ben Finegold's advice) wins! This prepares f5 by using up one of the pawn’s tempi, meaning that after f3 f5, you’ve still got a pawn on g7 so you maintain as many tempi as your opponent, the win is then trivial

Why does this matter? Well, I think it's not uncommon to have a three v three on the kingside. And although it's rarer, having both kings locked up on the queenside isn't exceptional. So being able to know whose winning can help for planning, you can avoid a lot of calculation when you're choosing to trade or not into a king and pawn endgame. Also, you now know both the right conversion if you're winning and the idea that'll best test your opponent if you're losing.