user13147: r2qkb1r/5ppp/p1bppn2/1p3P2/4P3/1BN5/PPP3PP/R1BQ1RK1 b qk - 0 1

1... b4 2. fxe6! bxc3 3. exf7+

In the above game, black played 1...b4 to encumber the white knight's journey to d5. Ambitiously, white sacrificed it with 2. fxe6! bxc3 3. exf7+: attacking the uncastled king, claiming the f7 square as his, and trapping the king on the center files. 
The inactivation of black's dark bishop justifies this sacrifice: it's not soon seeing daylight anytime. Similarly, the a8 rook is also passive; only the queen, light-squared bishop, and f6 knight appear to be able to defend a central onslaught. 
The black king can move to either Kd7 or Ke7, which aren't ideal in the least: these squares lie on two semi-open files available for white to take advantage of.
Altogether, this convinces white that black's life is going to be a terrible time. Still, the development of the queen's bishop must occur for placement of the connected rooks on d1, e1. He can choose squares such as g5, pinning the queen, or move his bishop along the a3-f8 diagonal after a capture on c3.

White needn't checkmate, but he should return with, beyond question, a better position than black. What final position does white visualize that warrants the knight's goodbye?
There can be an overwhelming number of variations in realizing that final position: after 4. Re1, black has a number of responses. If this is the best move, how does white calmly reassure himself that all of its responses are inadequate? If it's not the best move, what is and why?

I am not looking for a final, deeply analyzed (e.g. best) variation, but a winning position and the most effective way to achieve it: the moves that should be played first, and why. A 3-ply variation is enough, but a 4-ply variation is perfect. Maybe I also missed some critical details about the board, and would welcome these observations, if any.

user3774: The gut feeling is that White is not developed enough. For example, after 2. ef Qb6 3. Kh1 cb 4. ef Kd7 White cannot protect the e4 pawn. And it is not so much the pawn, but the Black Knight at e4, with the Black Queen at b6 forming an immediate (smothered) mate threat. White is pretty much forced to 5. Re1 Ne4 6. Be3 Qb7, and now White has to worry about the weakness at g2 as well as the Knight fork at c3 after cb.
So unless I am missing something obvious, the sacrifice doesn't look sound.

user-1: 
If this is the best move, how does white calmly reassure himself that all of its responses are inadequate?

White doesn't see a forced advantage in every line. He sees, what he believes to be, good winning chances. To quote Spielmann in The Art of Sacrifice in Chess:

In real sacrifices the player gives up material but is unable to calculate the consequences with accuracy; he has to rely on intuition



What final position does white visualize that warrants the knight's goodbye?

I think you answered your own question. After 2. fxe6 bxc3 3. exf7+ white probably thought that a lead in development and 2 pawns for the piece (potentially 3 if bxc3 is played) is sufficient compensation.
White doesn't have to look further than this if they are willing to take risks and figure out the attack 2 or 3 moves at a time. If white is risk averse, then they may not even consider 2. fxe6 bxc3 3. exf7+ at all.

user12540: Well I hope you are looking for a kind of algorithm or steps which would justify whether you should sacrifice or not. Irrespective of the above diagram which serves as an example I would like that you pay attention to the fact that "There is no perfect way to find out whether the sacrifice that you have played can be refuted or it will bring a complete victory".
You need to depend upon a few factors to bring out the best results. If you have ever watched videos on YouTube or see game annotations you may have heard that in some games the GM or the host says it is chess "intuition" which brings out this kind of magical moves. Chess intuition somewhat comes from experience or your own chess understanding. It can not be conscious calculations but looking at the board your active pieces vs. opponent's inactive pieces and with proper imagination can bring great results. If you see that after the sacrifice you can add fuel to the fire with every move, your king is safe and you can lodge a continuous pressure then a sac is definitely good for a winning attack. 
You need to take risks sometimes which you may win or lose depending upon some infinite chess factors which is not easy to state for every scenario. Even the great players make many mistakes and they forego the game. 

user10233: "I am not looking for a final, deeply analyzed (e.g. best) variation, but a winning position and the most effective way to achieve it:"
This statement is in conflict with itself. You are dealing with a position that requires CONCRETE analysis, i.e., accurate calculation leading to checkmate or a winning position. This is not a positional assessment where you are gaining a strong square, etc. The most effective way would be a final, deeply analyzed variation that is irrefutable.
In a blitz or rapid time control, I'd go in for it "because it looks good (intuitive senses). In a slow tournament game, I would settle into a long think and give it my "Karpov 7" treatment and assess it and compare it to my alternatives, which of course, is always how you should play. If I find my alternatives offer less, then I play it. Otherwise, I probably play safe. 

13147: <pre><code>r2qkb1r/5ppp/p1bppn2/1p3P2/4P3/1BN5/PPP3PP/R1BQ1RK1 b qk - 0 1

1... b4 2. fxe6! bxc3 3. exf7+
</code></pre>

<p>In the above game, black played 1...b4 to encumber the white knight's journey to d5. Ambitiously, white sacrificed it with 2. fxe6! bxc3 3. exf7+: attacking the uncastled king, claiming the f7 square as his, and trapping the king on the center files. </p>

<p>The inactivation of black's dark bishop justifies this sacrifice: it's not soon seeing daylight anytime. Similarly, the a8 rook is also passive; only the queen, light-squared bishop, and f6 knight appear to be able to defend a central onslaught. </p>

<p>The black king can move to either Kd7 or Ke7, which aren't ideal in the least: these squares lie on two semi-open files available for white to take advantage of.</p>

<p>Altogether, this convinces white that black's life is going to be a terrible time. Still, the development of the queen's bishop must occur for placement of the connected rooks on d1, e1. He can choose squares such as g5, pinning the queen, or move his bishop along the a3-f8 diagonal after a capture on c3.</p>

<ol>
<li>White needn't checkmate, but he should return with, beyond question, a better position than black. What final position does white visualize that warrants the knight's goodbye?</li>
<li>There can be an overwhelming number of variations in realizing that final position: after 4. Re1, black has a number of responses. If this is the best move, how does white calmly reassure himself that all of its responses are inadequate? If it's not the best move, what is and why?</li>
</ol>

<p>I am not looking for a final, deeply analyzed (e.g. best) variation, but <strong>a winning position and the most effective way to achieve it</strong>: the moves that should be played first, and why. A 3-ply variation is enough, but a 4-ply variation is perfect. Maybe I also missed some critical details about the board, and would welcome these observations, if any.</p>
 3774: <p>The gut feeling is that White is not developed enough. For example, after <code>2. ef Qb6 3. Kh1 cb 4. ef Kd7</code> White cannot protect the <code>e4</code> pawn. And it is not so much the pawn, but the Black Knight at <code>e4</code>, with the Black Queen at <code>b6</code> forming an immediate (smothered) mate threat. White is pretty much forced to <code>5. Re1 Ne4 6. Be3 Qb7</code>, and now White has to worry about the weakness at <code>g2</code> as well as the Knight fork at c3 after <code>cb</code>.</p>

<p>So unless I am missing something obvious, the sacrifice doesn't look sound.</p>
 -1: <blockquote>
  <p>If this is the best move, how does white calmly reassure himself that all of its responses are inadequate?</p>
</blockquote>

<p>White doesn't see a forced advantage in every line. He sees, what he believes to be, good winning chances. To quote Spielmann in <em>The Art of Sacrifice in Chess</em>:</p>

<blockquote>
  <p>In real sacrifices the player gives up material but is unable to calculate the consequences with accuracy; he has to rely on intuition</p>
</blockquote>

<hr>

<blockquote>
  <p>What final position does white visualize that warrants the knight's goodbye?</p>
</blockquote>

<p>I think you answered your own question. After 2. fxe6 bxc3 3. exf7+ white probably thought that a lead in development and 2 pawns for the piece (potentially 3 if bxc3 is played) is sufficient compensation.</p>

<p>White doesn't have to look further than this <em>if</em> they are willing to take risks and figure out the attack 2 or 3 moves at a time. If white is risk averse, then they may not even consider 2. fxe6 bxc3 3. exf7+ at all.</p>
 12540: <p>Well I hope you are looking for a kind of algorithm or steps which would justify whether you should sacrifice or not. Irrespective of the above diagram which serves as an example I would like that you pay attention to the fact that "There is no perfect way to find out whether the sacrifice that you have played can be refuted or it will bring a complete victory".</p>

<p>You need to depend upon a few factors to bring out the best results. If you have ever watched videos on YouTube or see game annotations you may have heard that in some games the GM or the host says it is chess "intuition" which brings out this kind of magical moves. Chess intuition somewhat comes from experience or your own chess understanding. It can not be conscious calculations but looking at the board your active pieces vs. opponent's inactive pieces and with proper imagination can bring great results. If you see that after the sacrifice you can add fuel to the fire with every move, your king is safe and you can lodge a continuous pressure then a sac is definitely good for a winning attack. </p>

<p>You need to take risks sometimes which you may win or lose depending upon some infinite chess factors which is not easy to state for every scenario. Even the great players make many mistakes and they forego the game. </p>
 10233: <p><strong>"I am not looking for a final, deeply analyzed (e.g. best) variation, but a winning position and the most effective way to achieve it:"</strong></p>

<p>This statement is in conflict with itself. You are dealing with a position that requires CONCRETE analysis, i.e., accurate calculation leading to checkmate or a winning position. This is not a positional assessment where you are gaining a strong square, etc. The most effective way would be a final, deeply analyzed variation that is irrefutable.</p>

<p>In a blitz or rapid time control, I'd go in for it "because it looks good (intuitive senses). In a slow tournament game, I would settle into a long think and give it my "Karpov 7" treatment and assess it and compare it to my alternatives, which of course, is always how you should play. If I find my alternatives offer less, then I play it. Otherwise, I probably play safe. </p>
