user15253: [FEN "8/8/8/3pk3/1K6/8/8/8 w - - 0 1"]

Even with White to move, the game is lost. OK, but according to what rule/principle?
With 1. Kc3 

I'm able to get first to one of the key squares (c3,d3,e3)
I get the diagonal opposition with the black King
I can occupy the black's promotion square before the black's King

In spite of all that, the ending is lost. Is it because I can't keep the oposition?
p.s.
The point of all this is to be able to evaluate this basic position only by applying rules, not having to calculate the concrete variations.

user12523: The diagonal (or horizontal) opposition doesn't do anything for you here because it doesn't help you prevent the black king from controlling the promotion square. If you slide down the c file, keeping the diagonal opposition, the black king just slide down the e file until it reaches e2, and then the pawn can simply march forward. And if you move your king to the d file, the black king can get the vertical opposition, forcing you to move out of the way until again it reaches e2 or c2, controlling the promotion square.
You could draw if you could get to the d file with the opposition, and without the pawn having extra tempos, but that would require black to blunder, for example 1.Kc3 Ke4 2.Kc2 Kd4?? 3.Kd2.
I guess the bottom line is that "not all oppositions are created equal", and often you do need to look at concrete variations anyway. Opposition is a tool, but what really matters here is getting control of the promotion square.

user7297: 
The point of all this is to be able to evaluate this basic position only by applying rules, not having to calculate the concrete variations.

Well, the basic rule of (single) pawn endings is that if the side with the pawn manages to get their king on one of the key squares (and the pawn can't be taken on the next move), it's a win.

I'm able to get first to one of the key squares

It doesn't matter that you're the first one to get there, that's probably where your mistake lies.

user14079: If you want to evaluate this ending with general rules, the rule of key squares is a good way to deal with it. However, this rule doesn't say "whoever reaches a key square first will get the result he is aiming for", it says "if the offensive king (here, Black) reaches one of the key squares, he will win."
Here after 1.Kc3 Ke4 2.Kd2 Kd4 you cannot prevent Black's king to reach one of the three key squares, so you lose.
[FEN "8/8/8/3pk3/1K6/8/8/8 w - - 0 1"]

1.Kc3 Ke4 2.Kd2 Kd4 3. Kc2 (3. Ke2 Kc3) Ke3


15253: <pre><code>[FEN "8/8/8/3pk3/1K6/8/8/8 w - - 0 1"]
</code></pre>

<p>Even with White to move, the game is lost. OK, but according to what rule/principle?</p>

<p>With 1. Kc3 </p>

<ul>
<li>I'm able to get first to one of the key squares (c3,d3,e3)</li>
<li>I get the diagonal opposition with the black King</li>
<li>I can occupy the black's promotion square before the black's King</li>
</ul>

<p>In spite of all that, the ending is lost. Is it because I can't keep the oposition?</p>

<p>p.s.
The point of all this is to be able to evaluate this basic position only by applying rules, not having to calculate the concrete variations.</p>
 12523: <p>The diagonal (or horizontal) opposition doesn't do anything for you here because it doesn't help you prevent the black king from controlling the promotion square. If you slide down the c file, keeping the diagonal opposition, the black king just slide down the e file until it reaches e2, and then the pawn can simply march forward. And if you move your king to the d file, the black king can get the vertical opposition, forcing you to move out of the way until again it reaches e2 or c2, controlling the promotion square.</p>

<p>You could draw if you could get to the d file with the opposition, and without the pawn having extra tempos, but that would require black to blunder, for example 1.Kc3 Ke4 2.Kc2 Kd4?? 3.Kd2.</p>

<p>I guess the bottom line is that "not all oppositions are created equal", and often you do need to look at concrete variations anyway. Opposition is a tool, but what really matters here is getting control of the promotion square.</p>
 7297: <blockquote>
  <p>The point of all this is to be able to evaluate this basic position only by applying rules, not having to calculate the concrete variations.</p>
</blockquote>

<p>Well, the basic rule of (single) pawn endings is that if the side with the pawn manages to get their king on one of the <a href="https://en.wikipedia.org/wiki/King_and_pawn_versus_king_endgame#Pawns_other_than_rook_pawns" rel="noreferrer">key squares</a> (and the pawn can't be taken on the next move), it's a win.</p>

<blockquote>
  <p>I'm able to get first to one of the key squares</p>
</blockquote>

<p>It doesn't matter that you're the first one to get there, that's probably where your mistake lies.</p>
 14079: <p>If you want to evaluate this ending with general rules, the rule of key squares is a good way to deal with it. However, this rule doesn't say "whoever reaches a key square first will get the result he is aiming for", it says "if the offensive king (here, Black) reaches one of the key squares, he will win."</p>

<p>Here after 1.Kc3 Ke4 2.Kd2 Kd4 you cannot prevent Black's king to reach one of the three key squares, so you lose.</p>

<pre><code>[FEN "8/8/8/3pk3/1K6/8/8/8 w - - 0 1"]

1.Kc3 Ke4 2.Kd2 Kd4 3. Kc2 (3. Ke2 Kc3) Ke3
</code></pre>
