Thanks for the example analysis, which I need to study more carefully later!

As to "local", I think you abuse the word.

My local assessment does not guarantee the local sente sequence and should not because there can be exceptions for the reverse sente to occur. I wrote principles for when to play locally as the creator or preventer.

As for yours, my definition does not guarantee the local sente sequence. What the difference for the go player?
With my defintion the go player knows that the position is sente in a rich environment and the go player knows that this rich environment is a good approximation of a real environment. As a consequence, without a 100% guarantee, the go player may really expect the local sente sequence.
With your defintion, when you claim a position is sente, how can you convinced a go player, without the help of a good approximation of the environment, that she may really expect a local sente sequence?

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . b . X . X X .
$$ | O O . O X . O a O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

When we discussed the position above we looked at the black options a and b. With only two options for black and only one option for white the tree seems quite simple isn't it?
Now what about the POSITION itself?
For me it is gote position.
For you, what is the status of the POSITION?

1) By the increasing local move values.

2) Even without knowing the largest or second-largest values of the environment, the principle inherent in the theorems is the one you already know: the sente player is expected to be able to play sente much earlier than the opponent should prevent it.

In a local endgame with gote and sente options, its status is



[14][22]

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . b . X . X X .
$$ | O O . O X . O a O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Oops I am not asking for the defintion of the status of a position but for the result of this defintion applied simply to the position above.
You warned us to distinguish clearly MGOTE_GOTE_OPTION from MGOTE_SENTE_OPTION and I am sure to be wrong when using your defintion comparaing MSENTE and MGOTE.
So Robert what is for you the status of the above position: local gote? local ambiguous? local sente?
If you prefer what are the values of MGOTE and MSENTE for applying your defintion to the position above?

See

https://www.lifein19x19.com/viewtopic.p ... 31#p280531

You refered here to a post dealing with your theorem:
"If F < T, usually start
- in the environment if MGOTE ≤ T,
- locally if MGOTE ≥ T (the creator chooses the gote option)."

In this post your are dealing with the status of OPTIONS and not the status of the POSITION.

Why not giving us (or remind us?) the result for the status of the POSITION? local gote? local ambiguous? local sente?

In that message, I calculated MSENTE and MGOTE so you have the info to compare both values.

Do you mean


But what is the conclusion?

In https://www.lifein19x19.com/viewtopic.p ... 12#p280612 you said in particular

while in https://www.lifein19x19.com/viewtopic.p ... 05#p280705 you said

Do you see why it is so difficult to understand what is your conclusion?
So again what is the status of the POSITION? local gote? local ambiguous? local sente?

When I wrire

"In a local endgame with gote and sente options, its status is

"
local gote := MSENTE < MGOTE

ambiguous := MSENTE = MGOTE

local sente := MSENTE > MGOTE
"

I mean a local endgame with gote and sente options.

When I write about a local endgame with one player:s follow-up and wtthout options that

"local gote :<=> MGOTE < MB,SENTE, MW,SENTE"

I mean a local endgame with one player:s follow-up

but I do not mean a local endgame with gote and sente options.

Different types of local endgames have different definitions of types!

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . b c X . X X .
$$ | O O . O X . O a O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Why not saying clearly your are unable to find the status of this position?
Analysing the options "a" and "b" is one thing but finding the status of the position is far more difficult.
When you write "local gote := MSENTE < MGOTE", though you did not clarify this point, I guess you take MGOTE = MGOTE_GOTE_OPTION but the difficulty is not really there. When you look for the status of the position above you have to take into account all options. Besides black "a" and black "b" how do you handle black "c"? Is black "c" dominated by black "a" or black "b"? If not what means MSENTE and MGOTE if you do not forgot the option black "c"?
Surely this position you proposed yourself is very interesting but with three options availbale for black it not an easy position. In addition, taking only option black c, can you determine easily if this option is gote or sente?

Because the opposite is true via application of my definition.



In this respect, you are right. This is an unanswered question, like all details of long sequences are unanswered and a detailed evaluation might give such answers. Privately, I did some such studies for this class of positions but proofreading demands more time, so this is unpublished.

That's a big the difference with my approach with a rich environment. Handling this black "c" option is for me obvious, with no supplementary count calculation.

"Obvious" is PR speech hiding the necessity of perfect global play for applying your definition.

No Robert your are wrong.
See for example my post https://lifein19x19.com/viewtopic.php?p=280640#p280640 where I used sequences that are not necessarily a perfect play. I called the strategy I used the "Always good exchange strategy".

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . b c X . X X .
$$ | O O . O X . O a O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Here again I can use this very efficient strategy with success, showing clearly that black c cannot be a good option in a rich environment.
BTW it is also obvious that black b cannot be either a good option.

In that post, you relied on equality of two counts after two sente sequences. Let me try to construct them for this example:

Click Here To Show Diagram Code

[go]$$B ordinary sente sequence
$$ ---------------------------
$$ | O . X X . 4 1 3 X 5 X X .
$$ | O O . O X 6 O 2 O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$B alleged alternative sente sequence
$$ ---------------------------
$$ | O . X X . 4 3 1 X 5 X X .
$$ | O O . O X 6 O 2 O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

For this sequence to make sense, it must be correct for White to reply locally and more specifically at 2.
However, why should this local reply be correct instead of a play elsewhere?

For possibly applying equality of two counts after two sente sequences, White replying locally at 2 must be correct. This must be proved before you can apply equality of two counts after two sente sequences.



I do not recall this your phrase. Where did you use it, what is the meaning and what does it tell us?



Why?



How do you show it at all and then even very clearly?



(The "either" confuses me.)

Do you presume your rich environment here?

If so, why cannot black b be a good option?

Why is it even obvious?

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . b c X . X X .
$$ | O O . O X . O a O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Oops a lot of questions Robert. Before presenting the "Always good exchange strategy" itself I think its interesting to show you how it works in practice on this example.

First of all I put this position P in a rich environment at high temperature T to form a global position G.
Then I look for the most promising local white and black moves.
For white, obviously white "c" is the best move.
For black it is not obvious but black "a" seems the more promising because this move allows to save the four black stones in the center.
Now play a game starting from position G. Each player will play in the environment until the temperature of the environment drops to a temperature t at which white will play at "c" or black will play at "a".
If white plays first at "c" then the score of the game is
S1 = -23.33 + t
If black plays first at "a" then the score of the game is
S2 = -12 - t
(in the calculation of the scores I deleted the term T/2 because it is present in each score and as such it is uninteresting)
Up to this stage nothing is really new. You find that the relevant temperature is when S1 = S2 => t = 5.66 => S1 = S2 = -17.66

Now I begin the "Always good exchange strategy".
For white I do not see another option than the move white "c" we considered => it remains only the black options black "b" and "c" and the question is the following: can black reach a score better than -17.66 by using options "b" or "c"?

Let's begin by black "b".
Now is the first point of the "Always good exchange strategy"

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . 1 . X . X X .
$$ | O O . O X . O 2 O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

I DECIDE to answer by without knowing if it is a good move or not (a tenuki could be better) because it is irrelevant in the "Always good exchange strategy".

From this position we reach the second point of the "Always good exchange strategy"

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . 4 1 3 X 5 X X .
$$ | O O . O X 6 O 2 O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

the following move may not be the best move (here again a tenuki might be better) but if white answers immediately by and then then I claim the sequence cannot be bad for black. In the startegy I used it is simply "good exchange" for black and it does not mean that the moves chosen are the best one!
Now is the conclusion:
After the all sequence cannot be bad for black => the score of the game after is an upper bound for the score of the game after followed by the best black and white moves. The score of the game after being S3 = -19 I conclude that with black cannot expect a score better than -19.
Because -19 < -17.66 then option "b" cannot be better than black "a".

If now you consider the option black "c" OBVIOUSLY you can consider the sequence equivalent to the previous one => black cannot expect a score better than -19 and option "c" cannot be better than black "a".

P is the local endgame (but not the 5.5 region).



For S1, White c takes -23 1/3, then Black (starts in the rest of the environment and) net takes t/2.

For S2, Black takes gote letting White have -12, then White (starts in the rest of the environment and) net takes -t/2.

S1 = -23 1/3 + t/2

S2 = -12 - t/2

To calculated t algebraically,

S1 = S2 <=>
-23 1/3 + t/2 = -12 - t/2 <=>
t = 11 1/3


(I need to read the rest of your text later.)

Yes OC I agree. It's better to consider instead the following diagram

Click Here To Show Diagram Code

[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . b c X . X .
$$ | O O . O X . O a O X X .
$$ | . O O O . O O X O X . .
$$ | . . . O O O . X O X . .
$$ | . . . . . O X X O X . .
$$ | . . . . . O O O O X . .
$$ | . . . . . . . . . . . . .[/go]

You are wrong Robert.

When you use the formula S1 = -23 1/3 + t/2 that means that you start from a position G1 made of position P, a rich environment at temperature t and with black to play.
And when you use the formula S2 = -12 - t/2 that means that you start from another position G2 made of position P, a rich environment at temperature t and with white to play.

Read again my approach: I consider only ONE position G made of position P and a rich anvironment at a high temperature T (T >> t) with black to play. I guess in your method you define two games at each node looking like a stop (in CGT) but in my method I use only ONE game.
Where is the difference with your approach?
For me, when I start from position G I do not know if white or black will play first in P and I have to study the two scenarios but without creating two new games.
In my previous posts I insisted on this point and I even show you that in the score you have also the term T/2. I said that this term was present in each score so that I can delete it but you cannot ignore this T/2 term was there!
More precisely let's assume black choses to play localy in P at temperature t. In that case black has obviously already gained something in the environment while temperature dropped from T to t. The gain for black is T/2 - t/2. If after black move in P black gets locally the score -12 and white is able to play first in the remaining environment then the final score is -12 + (T/2 - t/2) - t/2 = -12 + T/2 - t.
When you study my approach try to ignore your own method otherwise you will mixe the two approaches with a completly wrong result.