user17583: Prove that it is possible, from any starting KBN-K position, to accomplish the mate by manoeuvering to arrive at key positions from a well-known method or strategy whilst the losing king actively tries to avoid reaching the key positions that would lead to mate within 50 moves. In terms of the players' goals, the winning player's goal is to reach the key positions and still have moves to mate within 50 moves, and the losing player's goal is to avoid getting mated within 50 moves.
Given a starting KBN-K position, there exist paths through the tree of variations satisfying the winning player's goal with optimal play by the losing player. Is this satisfied for ALL legal KBNK positions using that particular stratagem only? How would one go about attempting to prove this using KBN-K tablebases?

user924: With the tablebase you can prove that all positions that can be won, can be won within 50 moves. But they're not designed to guarantee that certain "key positions" are visited in between (and if you pick them badly, then there are positions that can't be won in time anymore).
So you'd need to get your list of key positions and note in how many moves checkmate can be forced from there, using the existing tablebases.
And, for each key positions, you need to create new tablebases where the goal is not to checkmate, but to reach that key position.
Then for each KNB v K position, there needs to be at least one key position in the list for which the minimum moves to force it plus the minimum moves to checkmate from there is 50 or below.

user825: 
Is this satisfied for ALL legal KBNK positions using that particular stratagem only?

Absolutely not! Please don't try to prove something that's impossible. Consider this position.
[startflipped ""]
[FEN "3kB3/4N3/8/8/8/8/8/3K4 b - - 0 1"]


17583: <p>Prove that it is possible, from any starting KBN-K position, to accomplish the mate by manoeuvering to arrive at key positions from a well-known method or strategy whilst the losing king actively tries to avoid reaching the key positions that would lead to mate within 50 moves. In terms of the players' goals, the winning player's goal is to reach the key positions and still have moves to mate within 50 moves, and the losing player's goal is to avoid getting mated within 50 moves.</p>
<p>Given a starting KBN-K position, there exist paths through the tree of variations satisfying the winning player's goal with optimal play by the losing player. Is this satisfied for ALL legal KBNK positions using that particular stratagem only? How would one go about attempting to prove this using KBN-K tablebases?</p>
 924: <p>With the tablebase you can prove that all positions that can be won, can be won within 50 moves. But they're not designed to guarantee that certain "key positions" are visited in between (and if you pick them badly, then there are positions that can't be won in time anymore).</p>

<p>So you'd need to get your list of key positions and note in how many moves checkmate can be forced from there, using the existing tablebases.</p>

<p><em>And</em>, for each key positions, you need to create <em>new</em> tablebases where the goal is not to checkmate, but to reach that key position.</p>

<p>Then for each KNB v K position, there needs to be at least one key position in the list for which the minimum moves to force it plus the minimum moves to checkmate from there is 50 or below.</p>
 825: <blockquote>
<p>Is this satisfied for ALL legal KBNK positions using that particular stratagem only?</p>
</blockquote>
<p>Absolutely not! Please don't try to prove something that's impossible. Consider this position.</p>
<pre><code>[startflipped &quot;&quot;]
[FEN &quot;3kB3/4N3/8/8/8/8/8/3K4 b - - 0 1&quot;]
</code></pre>
