Clearly, e5 and f7 are corresponding: the first person to move on these squares loses
I don't think your definition of corresponding squares is quite correct. If the black king goes to f7, black does not necessarily lose.
In this position the fight is about the square f6. If black manages to get the king to f6 (with the white king not on f4) he can play g5 and create a passed pawn on the h file.
In this sense, e5 and f7 are corresponding because:
- If the white king goes to e5 first, black will play Kf7 putting white in zugzwang. White has to retreat with the king (or play b6 which is equally bad). Assuming white retreats the king from e5, you note another theme of corresponding squares: f6 and f4: if the black king goes to f6, white - in order to keep equality - has to reply with Kf4. Conversely, if white plays Kf4 himself first, black can reply with Kf6 putting white in Zugzwang. However you can easily see that black will always get to f6 favourably. For example 1. Ke5 Kf7 2. Ke4 (staying close to the square f4) Ke6! and white has to play Kf4 (which is answered with Kf6) or retreat further to the third rank, letting the black king to the fifth rank. In this case black is winning.
- If the black king goes to f7 first, white has to prevent Kf6 by playing Ke5. The position is still a draw in this case.
You can confirm for yourself that white is too slow to win on the queenside, i.e. capturing the pawn on a7 with the king and promoting the b pawn will take longer than black getting a queen on h1 (after playing g5...).
This means that white always has to keep an eye on the kingside pawns, because black is threatening to create a passed pawn with g5, hxg5 h4. Particularly this means, that the white king has to be able to reach the square ("pawn promotion square rule" for black pawn on h4): d4-d1-h1-h4. Particularly this means that the white king may never move to the sixth rank. So e6 is never going to be a corresponding square for the white king.