Diagram 1
White to move
How should White continue?

Black is playing for stalemate. In order to gain time for the king to approach, White must prevent the g-pawn from advancing two squares. That is why 1 Kb6? g5!= is inaccurate.
1...Kh2 2 Kb6 h3 3 Kc5 Kh1 4 Kd4 h2 5 Ke3 g6 6 Rg3 g5 7 Kf2 g4 8 Ra3 g3+ 9 Kxg3 Kg1 10 Ra1#

Black only draws with 111...Ke4? 112 Kf6(f7) Rf1+ 113.Ke6 Ra1 (113...Rg1 114 Kf6 Kd5 115 g6 Kd6 116 g7=)
114 g6 Ra6+ 115 Kf7 Kf5 116 g7 Ra7+ 117 Kf8 (117 Kg8! Kg6 118 Kh8=) 117...Kf6 118 g8N+.

Now White’s king is unable to “shoulder-block” its Black colleague. For this purpose, even a tempo is not too great a sacrifice!

Diagram 3
White to move
Can White win?

With accurate defense, the position is drawn.

In the game, Black erred with 3...Ra1?, and after 4 Rc8! (but not 4 Re8? Ra6+! =) 4...Ra6+ 5 Rc6 Rxa7 6 Rc7+ Rxc7 7 Kxc7, White controlled the distant opposition, which was then easily converted to the close opposition: 7...Kh7 8 Kd7! Kg6 9 Ke6 Kg7 10 Ke7 Kg6 11 Kf8, and Black resigned.

4 Ke6 Ra1 5 Rd8 (5 Re8 Ra6+! 6 Kf5 Rxa7 =) 5...Rxa7 6 Rd7+ Rxd7 7 Kxd7 Kh7!=. Black is saved only by the distant opposition - and not by the close opposition: 7...Kf7? 8 Kd6 is zugzwang, and it is White who takes the opposition.

The Black rook stands behind the passed h-pawn; but since White’s king has already arrived to blockade it, the pawn can only be employed as a distracting force. Black must initiate active operations on the opposite wing.
31...Ke6?! is useless after 32 Re1+ (Black cannot allow the White rook to reach the 7th). And after 31...Rh7?! 32 Kh3, it’s not easy to make further progress.