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Absolutely, unquestionably a theoretical dead draw, regardless of who's move it is. The only thing two things Black needs to know is to move the King directly backwards and then oppose the enemy king when it moves forward, and whenever he can, move his king to directly in front of the pawn. A line might be:
1. Kf4 Kg7! - moving the king straight back. Not 1...Kh5?? when 2. Kf5 wins.
2. Kf5 Kf7! - Opposing.
3. Ke4 Kg6! - Moving directly in front of the pawn.
4. Kf4 Kg7
5. Kf5 Kf7
6. g6+ Kg7
7. Kg5 Kg8
8. Kh6 Kh8
9. g7+ Kg8
10. Kg6 with a stalemate.
The key here is to not let white get the opposition with his king on the 6th rank. For white to win, he needs to be a square directly in front of the pawn with the opposition, or two without.
If the white king is on the same rank as the pawn and black can prevent white from getting into the above positions, it's a draw.
Yeah, I just went through it again myself. I realized that I never fully considered the simple option of the black king retreating, as opposed to moving side to side. That would explain how I missed it.
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