Thermography

[Bill Spight]

$$W White to play
$$ -----------------
$$ | d d b a b d O . . |
$$ | d c b d X O O . . |
$$ | b X X X X O . . . |
$$ | O X O O O O . . . |
$$ | d O O . . . . . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$W White to play
$$ -----------------
$$ | d d b a b d O . . |
$$ | d c b d X O O . . |
$$ | b X X X X O . . . |
$$ | O X O O O O . . . |
$$ | d O O . . . . . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------[/go]

Very good Bill.

If I am not wrong the result (area counting) for each move are the following:
move at "a" point : score -7
move at "b" points : score -5
move at "c" points : score -3
move at "d" points : score -1
tenuki : score +1

BTW, to adjust area score results for comparison with territory scoring, I add to them the number of White stones minus the number of Black stones in the original position. In this position there are 13 White stones and 6 Black stones, so we add 7.

The adjusted scores are then 0 for move a, 2 for move b, etc.

[Gérard TAILLE]

BTW, to adjust area score results for comparison with territory scoring, I add to them the number of White stones minus the number of Black stones in the original position. In this position there are 13 White stones and 6 Black stones, so we add 7.

The adjusted scores are then 0 for move a, 2 for move b, etc.

Oops I do not understand Bill. You are very aware that lots of sequences lead to a seki (your own two proposals are good example). In such case only the number of stones captured during the sequence are relevant and that does not look to fit your formula, does it?

[Bill Spight]

BTW, to adjust area score results for comparison with territory scoring, I add to them the number of White stones minus the number of Black stones in the original position. In this position there are 13 White stones and 6 Black stones, so we add 7.

The adjusted scores are then 0 for move a, 2 for move b, etc.

Oops I do not understand Bill. You are very aware that lots of sequences lead to a seki (your own two proposals are good example). In such case only the number of stones captured during the sequence are relevant and that does not look to fit your formula, does it?

It's not a conversion, it is for making comparisons. Of the two sequences I gave, the one starting with D-09 gets a territory result of +1, not 0, while one starting with E-09 gets a territory score of +2, which agrees with the adjusted score. Typically the adjusted score will be within 1 point of the territory score, depending on the specific rules.

[Gérard TAILLE]

BTW, to adjust area score results for comparison with territory scoring, I add to them the number of White stones minus the number of Black stones in the original position. In this position there are 13 White stones and 6 Black stones, so we add 7.

The adjusted scores are then 0 for move a, 2 for move b, etc.

Oops I do not understand Bill. You are very aware that lots of sequences lead to a seki (your own two proposals are good example). In such case only the number of stones captured during the sequence are relevant and that does not look to fit your formula, does it?

It's not a conversion, it is for making comparisons. Of the two sequences I gave, the one starting with D-09 gets a territory result of +1, not 0, while one starting with E-09 gets a territory score of +2, which agrees with the adjusted score. Typically the adjusted score will be within 1 point of the territory score, depending on the specific rules.

Yes Bill. Precisely because the adjusted score will be within 1 point, unfortunatly, comparing two sequences in area or territory scoring may give different results:

$$W White to play
$$ -----------------
$$ | . . b a . . O . . |
$$ | . . . . X O O . . |
$$ | . X X X X O . . . |
$$ | O X O O O O . . . |
$$ | . O O . . . . . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$W White to play
$$ -----------------
$$ | . . b a . . O . . |
$$ | . . . . X O O . . |
$$ | . X X X X O . . . |
$$ | O X O O O O . . . |
$$ | . O O . . . . . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------[/go]

The sequences beginning with "a" or "b" in the above diagram leads to the same score (+1) in territory scoring. In area scoring the sequence beginning with "a" is two points better than the sequence beginning with "b". That the reason I chose area counting for this problem.

[Bill Spight]

$$W White to play
$$ -----------------
$$ | . . b a . . O . . |
$$ | . . . . X O O . . |
$$ | . X X X X O . . . |
$$ | O X O O O O . . . |
$$ | . O O . . . . . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$W White to play
$$ -----------------
$$ | . . b a . . O . . |
$$ | . . . . X O O . . |
$$ | . X X X X O . . . |
$$ | O X O O O O . . . |
$$ | . O O . . . . . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------[/go]

The sequences beginning with "a" or "b" in the above diagram leads to the same score (+1) in territory scoring. In area scoring the sequence beginning with "a" is two points better than the sequence beginning with "b". That the reason I chose area counting for this problem.

Sure. No problem.

[dhu163]

I am gradually reading through this thread and several others to see what I can learn.

I found GT's question in post 33 interesting. I think that it should be possible to say a bit more about which environments correspond to situations, but I don't know how (yet). Thermography deals with this question restricting to ideal environments.

Is it true that in all environments, in that diagram, kosumi is dominated by max(keima, monkey jump)?

Taking the question literally, my answer is:

If you are comparing the resulting positions, then
No
Proof with notation from Bill's post 41
(1) Suppose A,B,C are mutually incomparable games.
Then no, it is not true that for all environments
A+E is at most as good as the best of B+E and C+E.

Proof:
Choose E=-A.
Then 0<=B-A is false (1) so whoever plays next can win in B+E=B-A.
Similarly, whoever plays next can win in C+E=C-A

_
hence only (1) requires proof, and diagrams such as post 13 work towards that.
_

Even this result is a bit interesting. It seems that difference games are key to showing if a move is optimal AND also if a move is suboptimal. Difference games are king in the world of games + arbitrary environments. Probably if the difference games don't tell you one way or the other, then anything is possible. There is the caveat that Bill has pointed out that reverses allow you know if a move is forcing in the CGT sense (not miai counting), which means sometimes you say two options are equivalent despite one dominating the other because one only dominates in other in environments where neither move is optimal.


In the below position kosumi seems the only move on the left to draw. At the least, keima and monkey seem to lose.

$$Bc Can Black to play draw?
$$ --------------------------------
$$ | . . . . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Can Black to play draw?
$$ --------------------------------
$$ | . . . . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

NB: if Black could play on the right to draw, then the kosumi would be a reverse. Hmm, if black attaches, and white blocks on the right, then black blocks on the right and draws. If black attaches, white connects, black keima on left does seem to draw though. If black attaches, white ataris instead, then black plays a monkey jump on the left and draws.
In most other variations, if W gets sente to block on the left, normally black will not be able to draw.
If we replace kosumi with keima on the right instead, (edit: changed my mind, that seems to reverse too, but not by playing the attachment. Black has to play the crude push on the right at N6 instead. If white responds, then black plays keima on the left)
If we replace kosumi with monkey jump on right (edit: changed my mind, that seems to reverse too with the M7 attachment. If white responds, black plays monkey jump on left)

I suppose this explains why Bill was comparing the kosumi sente, keima sente, monkey jump sente rather than just the first moves.

However, based on the above, I have now realised that

Because it reverses, that means we can assume the kosumi is answered by the attachment. But now if the opponent continues with the block, this was no better than the original position, so the original kosumi wasn't optimal unless the player continues with a response.
The only moves for white to continue are connect and atari, but these are dominated by the keima and monkey jump respectively as GT has argued.

Hence, I do in fact accept GT's argument that as moves, kosumi is dominated by keima/monkey jump here. That's fascinating.

__
The general argument format is:
Using convention LEFT wants to minimise, RIGHT wants to maximise.
G={T|A,B,C}
then if A={{T|D,E}|S} where D<=B and E<=C, then A's LEFT follower <=G. Hence A reverses, and we can replace A with the RIGHT options of A's LEFT follower, namely {D,E}.

But D is dominated by B, E dominated by C, so in all environments where play in G is optimal, a move to A is dominated by a move to B or dominated a move to C.

__
I am unable to say when A and (B,C) are incomparable in this setup

What went wrong? Can this be fixed? (this was written before I realised the above)

Where do GT's arguments go wrong? Probably from Bill and GT both assuming in diagrams that the first move is sente, though Bill repeatedly reminds that this is a dangerous assumption.
GT seems to assume alternating play locally (only valid if temperature of environment very low)

Using {black followers|white followers} notation,

post 38:
As a proof technique,
GT tries to say A={S|{G|H},{J|K}}, B={S|?}, C={S|?}
(not quite the same for the monkey jump but similar. The above is simpler. A, B,C represent the kosumi, keima and monkey jump positions)
He can prove B-G>=0 and C-J>=0 (but B and J are incomparable and so are C and G).

Hence in environment A+E and B+E,
-if best play in A+E moves to G+E, then B+E dominates this.
-if best play in A+E moves to J+E, then C+E dominates this.

This works if the environment E is colder than A for the first two moves. In such a case, I actually accept GT's proof technique as valid, and perhaps the first step in a series expansion. Obviously -A is as hot as A, so this is why E=-A breaks it. I don't know what can be done in general.

Unfortunately, GT is unable to say what happens if best play in A+E is to play in E on one of the first two moves in A+E.
-if best play in A+E moves to A+E', even if it then moves to S+E', how to find a strategy for black in B+E?
For example, what if white moves first in B+E, how should black respond?

Because the kosumi doesn't reverse, there are environments where the kosumi is optimal but it is not optimal to respond to it. And similar for the followup to the kosumi. Admittedly there are very few of these (i.e. the temperature is close to constant), which is why GT's idea is pretty good most of the time.
__
I suggest a series expansion above. I'm not sure of how to do the notation though as it is already fiddly.

post 207 Is there a nonzero solution to G+G+G=0?
(edit: removed lots of nonsense)
I'm guessing an infinite game is required.
For example, as a Go player who knows CGT only to a shallow level,
a ko should work. G = {-a|H}, H={G|2a}.

[Gérard TAILLE]

I am gradually reading through this thread and several others to see what I can learn.

I found GT's question in post 33 interesting. I think that it should be possible to say a bit more about which environments correspond to situations, but I don't know how (yet). Thermography deals with this question restricting to ideal environments.

Is it true that in all environments, in that diagram, kosumi is dominated by max(keima, monkey jump)?

Taking the question literally, my answer is:

If you are comparing the resulting positions, then
No
Proof with notation from Bill's post 41
(1) Suppose A,B,C are mutually incomparable games.
Then no, it is not true that for all environments
A+E is at most as good as the best of B+E and C+E.

Proof:
Choose E=-A.
Then 0<=B-A is false (1) so whoever plays next can win in B+E=B-A.
Similarly, whoever plays next can win in C+E=C-A

_
hence only (1) requires proof, and diagrams such as post 13 work towards that.
_

Even this result is a bit interesting. It seems that difference games are key to showing if a move is optimal AND also if a move is suboptimal. Difference games are king in the world of games + arbitrary environments. Probably if the difference games don't tell you one way or the other, then anything is possible. There is the caveat that Bill has pointed out that reverses allow you know if a move is forcing in the CGT sense (not miai counting), which means sometimes you say two options are equivalent despite one dominating the other because one only dominates in other in environments where neither move is optimal.


In the below position kosumi seems the only move on the left to draw. At the least, keima and monkey seem to lose.

$$Bc Can Black to play draw?
$$ --------------------------------
$$ | . . . . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Can Black to play draw?
$$ --------------------------------
$$ | . . . . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

NB: if Black could play on the right to draw, then the kosumi would be a reverse. Hmm, if black attaches, and white blocks on the right, then black blocks on the right and draws. If black attaches, white connects, black keima on left does seem to draw though. If black attaches, white ataris instead, then black plays a monkey jump on the left and draws.
In most other variations, if W gets sente to block on the left, normally black will not be able to draw.
If we replace kosumi with keima on the right instead, (edit: changed my mind, that seems to reverse too, but not by playing the attachment. Black has to play the crude push on the right at N6 instead. If white responds, then black plays keima on the left)
If we replace kosumi with monkey jump on right (edit: changed my mind, that seems to reverse too with the M7 attachment. If white responds, black plays monkey jump on left)

I suppose this explains why Bill was comparing the kosumi sente, keima sente, monkey jump sente rather than just the first moves.

I like your answer which is very interesting.

My question was the following : does it exist a non-ko environment for which kosumi is strictly better than both keima and monkey jump?

I am not sure your example is really valid because:

$$Bc Can Black to play draw?
$$ --------------------------------
$$ | . . . 5 3 6 O | X . 8 1 O . . |
$$ | X X . . 4 . O | X . . 2 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Can Black to play draw?
$$ --------------------------------
$$ | . . . 5 3 6 O | X . 8 1 O . . |
$$ | X X . . 4 . O | X . . 2 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bcm9 Can Black to play draw?
$$ --------------------------------
$$ | . . 5 X X O O | X . O 3 O 4 . |
$$ | X X 2 . O . O | X . 1 O X O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bcm9 Can Black to play draw?
$$ --------------------------------
$$ | . . 5 X X O O | X . O 3 O 4 . |
$$ | X X 2 . O . O | X . 1 O X O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

and white cannot win this game without winning the ko at which is in the environment ... though I was asking for a non-ko environment.

I am not convinced to have played the best white moves. Maybe you will find another sequence in which white will not be forced to win a ko in the environment. That is the point.

NB: if Black could play on the right to draw, then the kosumi would be a reverse. Hmm, if black attaches, and white blocks on the right, then black blocks on the right and draws. If black attaches, white connects, black keima on left does seem to draw though. If black attaches, white ataris instead, then black plays a monkey jump on the left and draws.
In most other variations, if W gets sente to block on the left, normally black will not be able to draw.
If we replace kosumi with keima on the right instead, (edit: changed my mind, that seems to reverse too, but not by playing the attachment. Black has to play the crude push on the right at N6 instead. If white responds, then black plays keima on the left)

I do not understand this NB. Could you please explain with diagrams?

[Bill Spight]

Using convention LEFT wants to minimise, RIGHT wants to maximise.
G={T|A,B,C}

What convention is that? The usual Cartesian convention?

Conway, for whatever reason, reversed the order of the Cartesian lateral axis for CGT, so that values increase as you move left. In CGT Black corresponds to the Left player and White corresponds to the Right player, and Black tries to maximize her scores.

Now that Conway, Berlekamp, and Guy have all passed away, perhaps, in time the lateral axis in CGT will revert to the usual Cartesian order, but I don't think that time has yet arrived.

[dhu163]

I'm not sure how helpful it is to argue that you ban ko takes for a "non-ko" environment, because that changes the rules of the game. So your would be the first invalid move. But I guess your logic is consistent if you ban the opponent from taking kos. I'm not too clear what "non-ko" environment means myself.

My point was that on the left diagram, (call the right one the environment) the kosumi move is strictly better than the keima as well as the monkey jump (this is implied if kosumi and keima are incomparable and kosumi and monkey jump are incomparable). Hence, this is a NO answer to the question (if only on positions)

Proof:

$$Bc Black to play can draw by playing kosumi (miai)
$$ --------------------------------
$$ | . . 1 . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing kosumi (miai)
$$ --------------------------------
$$ | . . 1 . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 7 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 5 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 7 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 5 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 8 O | X . . 9 O . . |
$$ | X X 4 3 6 . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 8 O | X . . 9 O . . |
$$ | X X 4 3 6 . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

Note that at is not ko due to shortage of liberties, atari, connect and die.

$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 0 O | X . . 7 O . . |
$$ | X X 4 3 6 . O | X . . . 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 0 O | X . . 7 O . . |
$$ | X X 4 3 6 . O | X . . . 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

at

However, in my second hide, I realise that all 3 moves are reverses, so in practice, I prove that it is impossible for the kosumi to be the only best move globally (i.e. locally and environment), which is like a YES answer to the question. (assuming all the difference games are correct) So I guess kosumi <= max(keima, monkey jump, environment best move) is the correct statement.

(NB In your diagram, I think is strange but ok, is a mistake unless B has enough ko threats, can draw by playing at . But I agree that with , the best score is a draw. see below)

The reason is that your actually does draw (if I read it right). This means that kosumi can always be answered with the attachment at M7 without black losing anything.
Proof:

W only has 3 possible moves. This is pretty much your original argument

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 4 . . . . O | X . . 1 O 5 . |
$$ | X X 2 . . . O | X . . . 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 4 . . . . O | X . . 1 O 5 . |
$$ | X X 2 . . . O | X . . . 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

4 and 5 were miai.

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 9 5 3 4 . O | X . . 1 O . . |
$$ | X X 8 6 . . O | X . . 7 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 9 5 3 4 . O | X . . 1 O . . |
$$ | X X 8 6 . . O | X . . 7 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . 0 1 O . . |
$$ | X X 6 5 8 . O | X . a 2 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . 0 1 O . . |
$$ | X X 6 5 8 . O | X . a 2 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

at a for miai.
playing 4 at 8 also draws I think.

Keima reverses Proof

White has two responses

$$Bc Black to play can draw by playing on the right =>Keima reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . . O 3 5 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Keima reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . . O 3 5 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>Keima reverses
$$ --------------------------------
$$ | . . 5 3 6 . O | X . 7 O 2 . . |
$$ | X X 4 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Keima reverses
$$ --------------------------------
$$ | . . 5 3 6 . O | X . 7 O 2 . . |
$$ | X X 4 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

This is miai

Monkey jump reverses Proof

White has two responses

$$Bc Black to play can draw by playing on the right =>monkey jump reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . O . . 3 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>monkey jump reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . O . . 3 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>monkey jump reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . O 9 2 . |
$$ | X X 6 5 8 . O | X . a 0 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>monkey jump reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . O 9 2 . |
$$ | X X 6 5 8 . O | X . a 0 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

This is miai after at a.

NB: this does not hold for the general 1st line problem, because normally the keima doesn't reverse and only gains. It only reverses here because the J7 is so close by, allowing the crude push to be a viable move.

[Bill Spight]

I'm not sure how helpful it is to argue that you ban ko takes for a "non-ko" environment, because that changes the rules of the game. So your would be the first invalid move. But I guess your logic is consistent if you ban the opponent from taking kos. I'm not too clear what "non-ko" environment means myself.

I think think that we can have kos if there is no ko fight. For instance, in this sequence,

$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

If does not fill the ko and takes it, White is not allowed to take the ko back. Somewhat artificial, but there you go.

Another possibility is that whoever takes a ko is komaster, which means that on their next turn they have to prevent the ko from being retaken. So if takes the ko, must win it.

AFAIK, the implications of difference games have only been proven for non-ko positions.

[Gérard TAILLE]

I'm not sure how helpful it is to argue that you ban ko takes for a "non-ko" environment, because that changes the rules of the game. So your would be the first invalid move. But I guess your logic is consistent if you ban the opponent from taking kos. I'm not too clear what "non-ko" environment means myself.

It seems the misunderstanding is here. Look the link giving what a difference game is : https://senseis.xmp.net/?DifferenceGame.
You will find the following article:
Ko caveat
Go is not strictly a combinatorial game because of kos. So difference games involving kos may not behave according to theory. Also, it may be right to make the play that the difference game says is wrong because it produces more or bigger ko threats for you or fewer or smaller threats for your opponent.

My view is the following : playing a difference game, even with ko, may be helpful for understanding a position and the various moves but, if you find a ko then you cannot claim for a reliable conclusion.
As a consequence you cannot say kosumi, keima, monkey jump are imcomparable according to difference games.

Let me reformulate my question in order to avoid any ambiguity.
I showed you a position in which black keima is the best move in a non-ko environment and
I showed you another position in which black monkey jump is the best move in another non-ko environment.

What about black kosumi? Here is my reformulated question :
Take as constraint that your are not allowed to use the kosumi move. Can you build in a non-ko environment in order to build a position for which you cannot reach the best score due to this constraint?

[Gérard TAILLE]

I think think that we can have kos if there is no ko fight. For instance, in this sequence,

$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

I believes your sequence is not the best one for black. It is possible to reduce the white score to W+1

$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 5 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 5 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

If does not fill the ko and takes it, White is not allowed to take the ko back. Somewhat artificial, but there you go.

Another possibility is that whoever takes a ko is komaster, which means that on their next turn they have to prevent the ko from being retaken. So if takes the ko, must win it.

AFAIK, the implications of difference games have only been proven for non-ko positions.

I have another suggestion. When playing a difference game consider always that the one who is playing first is komaster. That way, if you do not manage to win though you are komaster this result seems reliable.

[Gérard TAILLE]

I'm not sure how helpful it is to argue that you ban ko takes for a "non-ko" environment, because that changes the rules of the game. So your would be the first invalid move. But I guess your logic is consistent if you ban the opponent from taking kos. I'm not too clear what "non-ko" environment means myself.

My point was that on the left diagram, (call the right one the environment) the kosumi move is strictly better than the keima as well as the monkey jump (this is implied if kosumi and keima are incomparable and kosumi and monkey jump are incomparable). Hence, this is a NO answer to the question (if only on positions)

Proof:

$$Bc Black to play can draw by playing kosumi (miai)
$$ --------------------------------
$$ | . . 1 . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing kosumi (miai)
$$ --------------------------------
$$ | . . 1 . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 7 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 5 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 7 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 5 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 8 O | X . . 9 O . . |
$$ | X X 4 3 6 . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 8 O | X . . 9 O . . |
$$ | X X 4 3 6 . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

Note that at is not ko due to shortage of liberties, atari, connect and die.

$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 0 O | X . . 7 O . . |
$$ | X X 4 3 6 . O | X . . . 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 0 O | X . . 7 O . . |
$$ | X X 4 3 6 . O | X . . . 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

at

However, in my second hide, I realise that all 3 moves are reverses, so in practice, I prove that it is impossible for the kosumi to be the only best move globally (i.e. locally and environment), which is like a YES answer to the question. (assuming all the difference games are correct) So I guess kosumi <= max(keima, monkey jump, environment best move) is the correct statement.

(NB In your diagram, I think is strange but ok, is a mistake unless B has enough ko threats, can draw by playing at . But I agree that with , the best score is a draw. see below)

The reason is that your actually does draw (if I read it right). This means that kosumi can always be answered with the attachment at M7 without black losing anything.
Proof:

W only has 3 possible moves. This is pretty much your original argument

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 4 . . . . O | X . . 1 O 5 . |
$$ | X X 2 . . . O | X . . . 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 4 . . . . O | X . . 1 O 5 . |
$$ | X X 2 . . . O | X . . . 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

4 and 5 were miai.

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 9 5 3 4 . O | X . . 1 O . . |
$$ | X X 8 6 . . O | X . . 7 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 9 5 3 4 . O | X . . 1 O . . |
$$ | X X 8 6 . . O | X . . 7 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . 0 1 O . . |
$$ | X X 6 5 8 . O | X . a 2 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . 0 1 O . . |
$$ | X X 6 5 8 . O | X . a 2 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

at a for miai.
playing 4 at 8 also draws I think.

Keima reverses Proof

White has two responses

$$Bc Black to play can draw by playing on the right =>Keima reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . . O 3 5 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Keima reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . . O 3 5 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>Keima reverses
$$ --------------------------------
$$ | . . 5 3 6 . O | X . 7 O 2 . . |
$$ | X X 4 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Keima reverses
$$ --------------------------------
$$ | . . 5 3 6 . O | X . 7 O 2 . . |
$$ | X X 4 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

This is miai

Monkey jump reverses Proof

White has two responses

$$Bc Black to play can draw by playing on the right =>monkey jump reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . O . . 3 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>monkey jump reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . O . . 3 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>monkey jump reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . O 9 2 . |
$$ | X X 6 5 8 . O | X . a 0 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>monkey jump reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . O 9 2 . |
$$ | X X 6 5 8 . O | X . a 0 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

This is miai after at a.

NB: this does not hold for the general 1st line problem, because normally the keima doesn't reverse and only gains. It only reverses here because the J7 is so close by, allowing the crude push to be a viable move.

$$B Can Black to play draw?
$$ --------------------------------
$$ | . . . . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$B Can Black to play draw?
$$ --------------------------------
$$ | . . . . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

The basic question is the following : can black play and draw in the diagram above.

$$B
$$ --------------------------------
$$ | . . . a 3 . O | X . . 1 O . . |
$$ | X X . . b . O | X . . 2 . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$B
$$ --------------------------------
$$ | . . . a 3 . O | X . . 1 O . . |
$$ | X X . . b . O | X . . 2 . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

I think we agree that the three moves above are the beginning of the best sequence.

But here our sequences differ: you play at "a" and you conclude to the draw. Fine but that means that you must find another white move to try to win doesn't you? For that purpose I suggested the "strange" move white at "b

$$Bc
$$ --------------------------------
$$ | . . . 5 X . O | X . . X O . . |
$$ | X X . . 4 . O | X . . O . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc
$$ --------------------------------
$$ | . . . 5 X . O | X . . X O . . |
$$ | X X . . 4 . O | X . . O . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

After at "b" I played and you react by saying : ":b5: is a mistake unless B has enough ko threats". Your are right but here is really the point: if instead you play another move then black will simply lose without ko! The is the only move avoiding the loss without condition. In that sense is the best move.

What conclusion? In the initial position the best result black can reach is a disadvantageous ko for a draw.
IOW if black is komaster on the environment then black manages to draw otherwise black will lose.

[Gérard TAILLE]

I'm not sure how helpful it is to argue that you ban ko takes for a "non-ko" environment, because that changes the rules of the game. So your would be the first invalid move. But I guess your logic is consistent if you ban the opponent from taking kos. I'm not too clear what "non-ko" environment means myself.

My point was that on the left diagram, (call the right one the environment) the kosumi move is strictly better than the keima as well as the monkey jump (this is implied if kosumi and keima are incomparable and kosumi and monkey jump are incomparable). Hence, this is a NO answer to the question (if only on positions)

Proof:

$$Bc Black to play can draw by playing kosumi (miai)
$$ --------------------------------
$$ | . . 1 . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing kosumi (miai)
$$ --------------------------------
$$ | . . 1 . . . O | X . . . O . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+2
$$ --------------------------------
$$ | . . 3 1 4 . O | X . 8 5 O . . |
$$ | X X 2 . . . O | X . 7 6 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 7 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 5 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the keima, W+1
$$ --------------------------------
$$ | . 7 3 1 4 . O | X . 9 8 O . . |
$$ | X X 2 6 . . O | X . . 5 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 8 O | X . . 9 O . . |
$$ | X X 4 3 6 . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 8 O | X . . 9 O . . |
$$ | X X 4 3 6 . O | X . . 7 0 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

Note that at is not ko due to shortage of liberties, atari, connect and die.

$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 0 O | X . . 7 O . . |
$$ | X X 4 3 6 . O | X . . . 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black loses with the monkey jump, W+1
$$ --------------------------------
$$ | . . 5 2 1 0 O | X . . 7 O . . |
$$ | X X 4 3 6 . O | X . . . 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

at

However, in my second hide, I realise that all 3 moves are reverses, so in practice, I prove that it is impossible for the kosumi to be the only best move globally (i.e. locally and environment), which is like a YES answer to the question. (assuming all the difference games are correct) So I guess kosumi <= max(keima, monkey jump, environment best move) is the correct statement.

(NB In your diagram, I think is strange but ok, is a mistake unless B has enough ko threats, can draw by playing at . But I agree that with , the best score is a draw. see below)

The reason is that your actually does draw (if I read it right). This means that kosumi can always be answered with the attachment at M7 without black losing anything.
Proof:

W only has 3 possible moves. This is pretty much your original argument

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 4 . . . . O | X . . 1 O 5 . |
$$ | X X 2 . . . O | X . . . 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 4 . . . . O | X . . 1 O 5 . |
$$ | X X 2 . . . O | X . . . 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

4 and 5 were miai.

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 9 5 3 4 . O | X . . 1 O . . |
$$ | X X 8 6 . . O | X . . 7 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . 9 5 3 4 . O | X . . 1 O . . |
$$ | X X 8 6 . . O | X . . 7 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . 0 1 O . . |
$$ | X X 6 5 8 . O | X . a 2 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Kosumi reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . 0 1 O . . |
$$ | X X 6 5 8 . O | X . a 2 9 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

at a for miai.
playing 4 at 8 also draws I think.

Keima reverses Proof

White has two responses

$$Bc Black to play can draw by playing on the right =>Keima reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . . O 3 5 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Keima reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . . O 3 5 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>Keima reverses
$$ --------------------------------
$$ | . . 5 3 6 . O | X . 7 O 2 . . |
$$ | X X 4 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>Keima reverses
$$ --------------------------------
$$ | . . 5 3 6 . O | X . 7 O 2 . . |
$$ | X X 4 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

This is miai

Monkey jump reverses Proof

White has two responses

$$Bc Black to play can draw by playing on the right =>monkey jump reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . O . . 3 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>monkey jump reverses. This one is B+1
$$ --------------------------------
$$ | . 4 . . . . O | X . O . . 3 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

$$Bc Black to play can draw by playing on the right =>monkey jump reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . O 9 2 . |
$$ | X X 6 5 8 . O | X . a 0 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------

Click Here To Show Diagram Code

[go]$$Bc Black to play can draw by playing on the right =>monkey jump reverses
$$ --------------------------------
$$ | . . 7 4 3 . O | X . O 9 2 . |
$$ | X X 6 5 8 . O | X . a 0 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

This is miai after at a.

NB: this does not hold for the general 1st line problem, because normally the keima doesn't reverse and only gains. It only reverses here because the J7 is so close by, allowing the crude push to be a viable move.

You use a lot of time the wording "reverse". What does it means exactly to you ? It looks not the definition you can find here:
https://senseis.xmp.net/?Reversible and, as a consequence, I am not sure to understand what you are really saying.

[dhu163]

Thanks for your explanations. They all make sense, especially the variation with on the monkey jump. That ko does seem to be critical, so my "proof" is flawed for missing it out.

I meant for "reverse" to be the same as the SL definition.

Instead of "if not" can black with the move win [the difference game], I tried to prove "can white on the previous move draw with a local response" which is equivalent IIUC. Though the colors in my diagram are switched relative to SL.