Right. White cannot typically afford to playGérard TAILLE wrote:OK Bill my understanding is the followingBill Spight wrote:I know I did some work on this position some years ago, because I commented on it on SL, at https://senseis.xmp.net/?L2GroupWithDescent#toc4 . But it doesn't ring a bell. There are a number of kos in the game tree, but I am not well, and I am not interested in doing a full analysis, at least not now.
However, a few days ago I found some interesting things with the following first 4 plays.
At this point White threatens to win the ko in 1 move. We assume that t > 1.
= t
Since Black played first we can leave it at that.
Result: -5½ + t
Or Black can kill the corner in 3 net moves.
,
,
= t
Result: 21 - 3t
The temperature of indifference occurs when
-5½ + t = 21 -3t, or when
t = 6⅝
We anticipate each play in the ko gaining 6½ points on average, not 6⅝. Verrry interesting.
More later.
if t > 6⅝ both players play in the environment
if 6½ < t < 6⅝ white will not play in the environment; black will play in the corner and white will not defend the corner:tenuki
tenuki
Finally if t < 6½ both player will play in the corner. If the case black plays in the corner then white will defend withtenuki
tenuki
in the corner until the temperature drops to 6½. 
But note the peculiarity after
., , = t, fills the koIf White allows Black to do so, Black can kill the corner in 3 net plays.
Result: 21 - 3t
, , , = t, , take koResult: -6 + 2t
The temperature of indifference occurs when
-6 + 2t = 21 - 3t, or when
t = 5.4
Really? Do we really have a 5 stage iterated ko? They exist, OC, but is this really one? Does White have to wait until t = 5.4 before playing
?No. Take a look at the position after
.At this point
is essentially a 1 point sente. 
Now gains only 1 point and the result is -5 + t, so the temperature of indifference occurs when-5 + t = 21 - 3t , or when
t = 6½
The thing is, White does not play
until Black has played . Now, Black does not have to wait until after
to play . Black could play there after or . The point is, though, that it is played at temperature 1. And since we are assuming t > 1, it need not be played at all in this sequence. White stops after . (We may show for clarity, OC.)Note that
gains, on average, 6½ points. It is not an inferior play, although futile. In practice, White will wait for Black to play at a before taking the ko to start with. More later.
, 
, 
= t
So, even though the mast rises vertically at s = 8, that is not the average territorial value at t = 4. That value is -5 + 2t = 3. 
(takes ko) looks strange and I hardly see a gain with this move.
the result is quite different. If now white plays in the environment black can kill the corner. I conclude 
in the corner and 
tenuki
= u, 
must still take the ko.
must win the ko.
= t