So I guess before answering that sort of thing it might be worth moving on slightly into Chapter 2.1 on fractions.
Basically he shows a bunch of corridor examples, which while undecided have fractional values. It’s summarized in the table of Figure 2.2. (Just ignore the life and death issues in these examples imagine they’re connected to a big live group if you want to)
The way the calculations work, is they look at what happens when white plays or black plays and then take an average. So for the smallest corridor, if Black plays 1 point for black, if white plays 0 points. The average is 1/2, and some sources will call this the expected amount of territory for Black here.
In the next corridor, if Black plays it’s B+2 but if white plays its the first corridor which is B+0.5, and so you again average these two to get 2+0.5/2 = 1.25 or 1 and 1/4. Then you keep going the average of 3 and 1.25 is 2.125 or 2 and 1/8 etc.
I think the one thing books generally do badly here is motivate this calculation. Some books, including a Japanese book I was reading will try to motivate it like “We don’t know who will play here next so it’s like a 50/50 chance either will play and so you average the two future outcomes”. I think Antti’s endgame book comes at it from a more useful or practical motivation, where it turns out that when you add the expected territories you actually wind up with a number which when played really is the territory that Black gets in the game (if it’s the whole game).
So the example is this Christmas tree problem:
when you add up the numbers you get that Black should expect 7 points of territory, and when you play it out correctly in a sense, then this actually does happen to be the case.
So in a way that’s a better motivation, but it’s still mysterious, and it doesn’t fully address what happens if you play the endgame “wrong”. Something myself and @Jon_Ko played about with was trying to see where when you play such an endgame wrong the extra point “appears from”. It’s kind of that you have to make a total mistake of over one point over the course of the sequence. It might not be one single move that loses you a point. Even some slightly wrong move orders (judging by point values) will still lead to B+7.
Anyway, the idea of value of a move is typically in these cases taken as the difference between the score after the move is played and the expected score of the position. So when white or black moves in the half point corridor the value moves up to 1 or down to 0, by 1/2 a point, so the value of the move is 1/2. If white or black moves on the 1+1/4 point corridor it either jumps up to 2 points or down to 1/2 point which means it’s worth 3/4 of a point and so on.
These are precisely the temperature values in the table of 2.4 of Mathematical Go. There however they’re presented as 1-1/2, 1-1/4 etc to show how they increasingly get closer to being worth 1 point but not exactly. Antti points out that the general formula for the value is (2^x-1)/2^x where I guess x is the number of points black can make in the corridor with one move, or the “length of the corridor”. The expected territory I guess is more like (x-1)+(1/2^x).
There’s a comment in Berlekamp about how a corridor that is open on both ends is kind of like a corridor that is one length shorter. The precise reasoning isn’t explained, but the Go players explanation is that both open ends are miai, and if white plays one black can play the other. That’s also reflected in the table.
I don’t want to jump too far ahead yet, so I can leave it there before one tries to read the headache of “Chilling” which is also in the books subtitle 