Beautiful Puzzles/Studies #5 - A Genius Endgame Save!

Intro

I came across a fascinating study in the endgame, which is shown below:

This position is White to move and continue in the best way possible. Avoid scrolling further if you want to think and try to find the continuation before I give it.

The Answer

Well, let's eliminate any chance of white winning this game.

By quickly using the rule of the square or just intuition, we can immediately tell that the white king, despite having the first move, cannot stop the black a pawn from promoting. Moreover, white's kingside "counterplay" (if you can call it that) is completely harmless because not only is the black a pawn much faster, but black's king can easily stop the pawn(s). Here we see quality over quantity.

And, if I am giving this as a puzzle, I think it is very clear that we are trying to find a way to save this for white. Have you found the only resource, yet an amazing one that saves half a point, that white has up their sleeve?

Notice the unusual configuration of the white pawns as well as the black pawn on g6. This should lead to ideas of trying to box in the white king and creating a self-stalemate situation. In a self stalemate, a player intentionally traps their own king behind a wall of their own pieces in order to intentionally create stalemate. It is a very clever way in endgames to save lost positions (#SLP).

White of course starts with Kg3 (I mean, what else?) and black pushes their pawn. Now, as opposed to trying to catch the pawn somehow, White runs the other way, to h4! We are starting to see the self-stalemate in action --- white is trying to box their king. Now, after black pushes a3, white seals the door shut with g3, and no matter what black plays, this is already a draw. Note how black's pawn on g6 helps white in blocking the escape route via h5.

It should be noted that had black's pawn already promoted on a1 before g3, then Qh8# is game over (I guess you could call that self-checkmate).

Conclusion

That's it! I hope this was a fun puzzle, simple yet intriguing at the same time! See you soon!