Wouldn't another possible conclusion be that

A) the second diagram is "normal" (even), because

B) in the first diagram was a mistake, but

C) White makes an equivalent mistake with , bringing it back to the "normal" position?

I think that if it can be proved that - in the first diagram are 100% good and is the only mistake, then it is necessary that white has also made an equally large mistake in the second diagram.

In other words, I think your logic is valid-- the difficulty is in proving that your data (that - are correct) is valid. I don't doubt that it is correct-- I just don't know how one could prove that rigorously (or disprove it, either-- I'll have a hard time believing that the enclosure is wrong, as I've suggested above).

Which leads me to my question: What did white do wrong in the second diagram? Is it , an approach that plays into a pincer?

I am not sure, but I think that permutation (changing the order) is what you are talking about now.

As far as logic goes, if White did something wrong in the first line of play, so that the end result is bad for White, White did something wrong in the second line of play, as well, since it produced the same result.

However, the usual case is one in which we do not know whether a result is good or bad. And in general, we know that the order of play matters, so that we cannot freely permute it. A play that might have been good when it was played may not be good when other stones have been played.

So we are talking about a heuristic, a guide for play and judgement that is not infallible.

Of course there are mathematicians who are far better than me, but I think there is one relatively easy explanation to it. For this discussion lets ignore the effects of ko or similar stuff.

I will model the problem as follows. For every position we can say something like, Black leads by 10 points or White leads by 20 points. So we can assign each position a score. Lets agree on a convention that if Black leads, the score is positive, and if White leads, the score is negative. So Black leading by 10 points means a score of +10, White leading by 20 points means a score of -20. If a position is even, then the score would be 0. An optimal move by either player does not change the score of the position. A sub-optimal move by Black decreases the score, and a sub-optimal move by White increases the score. Black alone cannot increase the score of the position, and White alone cannot decrease the score of the position.

In your first sequence, you claim that all the moves are (more or less) optimal for both players, except for . Hence, the score of the position after each move before should be about 0. But causes the score to be changed to +k, where k is a positive number much larger than 0. If you plot a graph of score against move number, then you see that the graph starts with score 0, is flat until , and then an increase up to +k.

Given any move reordering (sequence dissection) of the first position, such as your second sequence, the first point and the last point on this graph are the same since they are the same position. From here we can immediately conclude that there must be at least one White move that is sub-optimal, although we cannot tell whether there are sub-optimal Black moves. The reason is because somewhere along the way the score must increase.

I think this gives a somewhat intuitive idea. I think my modelling is right, but if it is not, maybe Bill or someone else can correct me.

As long as we merely permute a fixed sequence of moves, starting from the same position, with no stones being captured either way, the ending positions are also the same. As Unkx80 points out, the net score shift (e. g. a loss for White) over the first sequence then necessarily equals the net score shift over the second. We can say for sure that, if one color plays worse in one permutation than in the other, the opponent does so, too. This relation is generally valid, but without "Go knowledge", useless for practical analysis (for reasons illustrated by LukeNine45's three-legged dog).

I think that even more Go knowledge is required to sensibly perform the actual dissection, which goes beyond permuting. In principle, tewari, or "sequence dissection", can help to analyse a particular difficult to judge about subsequence (may consist of just a single move), if both the complement of that subsequence and the whole sequence are easier to judge about. Likewise, it can help to analyse an obscure whole sequence which can be devided into subsequences that are clearer. Both ways may also involve permutations.

Supposedly, the most popular application of this principle is to analyse a whole sequence by:

1. removing a well-chosen subset of its moves that is considered sufficiently close to neutral, to get a remainder which is easier to evaluate
2. evaluating that remainder

There is a catch in here: If the removed subset is biased, you can prove about anything! This might be (part of) what makes "ordinary amatEUr players often feel suspicious about pro's sequence dissection" (minue). The undone moves must be significantly easier to judge about than the whole sequence, so we are able to see that their net effect is really close to neutral, or the entire procedure is moot.