For most of the other theorems, such as related to gote and sente options, there is no requirement of no dominated and no reversible plays. That is because the proofs do not need such presuppositions. The double sente theorem needs the no reversible presupposition because it enables the proof for a more general values range also covering, as you have noticed, sekis and the like.

Mostly the theory of Bill Spight and me avoids CGT low level things, such as infinitesimals or preliminary tree simplifications. Usually, our theory is more accessible. Only sometimes we need CGT techniques.

Let me elaborate a bit more on this. At first glance, the theorem looks as if it was written without work because I have designed it as close to go players' thinking as possible. To achieve this, Bill had to work hard on the first theorem on local endgames with gote and sente options and then I did mathematical research on them for months writing and proving more fundamental theorems or preliminary propositions.

This starts, in particular, with the local move values MGOTE and MSENTE and the alternating sum ∆T of the environment's move values.

For every move sequence, one gets a term similar to M + C + ∆T, in which move values and a count are added. This required me to prove fundamental theorems for the validity of such a sum of move values and counts.

One compares such terms of every two move sequences while considering one player maximising and the opponent minimising. The purpose is to retrieve value comparisons in theorems, such as T ? MGOTE or 2∆T ? MSENTE for some appropriate relation '?'.

Other theorems establish equivalence of a) principles using move values and b) resulting counts of move sequences. (For other kinds of local endgames, there is also equivalence to using gains.)

Ω is an alternating sum for the tails of move sequences. ∆T|F is the alternating sum of the move values in the environment and the local follow-up move value F. T1 is the second-largest move value of the environment. The following term occurs in several propositions and proofs: ∆T1|F - ∆T1. Now, it was very useful for me to prove ∆T1|F - ∆T1 = Ω if the temperature is high. In a citation of the related proof, it is easier to see how work was done when writing out such in detail

"∆T1|F - ∆T1 = (T1 - T2 +...- TL + F - TL+1 + TL+2 -...) - (T1 - T2 +...- TL + TL+1 - TL+2 +...)"

then transforming it to Ω. This is for even L and a similar representation occurs for odd L.

Only as one of the last steps, the resulting counts C1, C2 and C3 are identified with the terms of move values for the move sequences. Before, several pages of detailed mathematical work on all the move values must be done.

While the theorem looks like copy and paste of an informal method, I established it by proper, detailed mathematical proving. That's why it is a theorem and not informal.


EDIT: high temperature

And what are the other sequences besides C1, C2, C3? I see only one sequence family C4 - if the creator starts in the environment and the preventer replies in the environment.

Yours is another sequence family indeed and Bill found its treatment in the proofs.

We as players and my maths rely on taking simple gotes without follow-ups in order of decreasing-or-constant move values so this rules out most sequences indeed. However, even so it may not be obvious that the remaining follow-up with move value F shall abide by the same assumption. In my proofs, I treat this meticulously so that, eventually, the sente option only needs one sequence with its one resulting count instead of two sequences with two parameters for counts.

Let me come back to a previous question. Assuming simply T = MGOTE, I was asking for only one example for which:
1) one player has a gote and a sente option
2) F < T

You answered with a position in which |T - MGOTE| = 1/6. Robert, sure this position is interesting but this position has nothing to do with my question because I assumed T = MGOTE.

Seeing you did not find any other position (or tree) I conclude that we cannot have simultaneously
1) T = MGOTE
2) One player has a gote and a sente option
3) F < T
That means that your theorem cannot be applied if T = MGOTE

When I saw this condition to apply your theorems for late endgame I immediatly understood that I will never apply your theorems for a very simple reason:

Click Here To Show Diagram Code

[go]$$
$$ -------------------
$$ . . . O . a X . . .
$$ . . . O O O X . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

For me, after a black move at "a", a follow-up exists because I use AGA rules! Because in practice it exists almost always such area in the environment your conditions for a late endgame are not fullfilled in practice.

I do not understand why you call this option a SENTE option.
After you play in the environment. That means this sequence is gote isn't it?
In that case how can you apply your theorem 128?

Non-existence is proved by a proof of non-existence.

Non-existence is not proved by referring to my limited time.

Right, these theorems are for territory scoring while area scoring during the late endgame requires more sophisticated maths, such as infinitesimals of CGT. Have fun always applying infinitesimals and full-blown CGT! ;)

Because, for an assessment as a local sente, my (and Bill's) related definition of an option is purely local and only considers it and the opponent's sequence.

Local means local. Local does not include anything outside the local region. Therefore, it is local. Local - not global!

(Bill has sometimes used a different assessement a la John Conway by considering arbitrarily many multiple copies of the local region, combined play in the union of all copies and forming the average count for the limit to infinity on the number of copies.)

(Global sente is a different concept. A local sente might, or might not, be a global sente (e.g., at different ambient temperatures).)


1 EDIT

imho
1) is sente in the local game but not in the whole game
2) if the whole game consist of N copies of the local game then would always be sente

What about this one?

Click Here To Show Diagram Code

[go]$$
$$ +-------------------------+
$$ | O . X X . . . . X . . X |
$$ | O O . O X . O . O X X X |
$$ | . O O O . O O X O X . X |
$$ | . O X 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]

Thank you for this attempt Dany.
If I count correctly: MGOTE = MSENTE = F = 5 (=>ambiguous)
Because I assumed T = MGOTE then F < T is not fulfilled.

Distinguish MGOTE_GOTE_OPTION from MGOTE_SENTE_OPTION!

What are your figures?

H = -11
MGOTE = (H - R) / 2 = (-11 -(-23))/2 = 6

S = -18
MSENTE = S - R = -18 - (-23) = 5

MGOTE_SENTE_OPTION = (-8 - (-18)) / 2 = 5 = MSENTE = F (ambiguous).

Theorem for the early endgame: F < T = MGOTE_GOTE_OPTION = 6 so either start in the environment or local with the gote option is correct.

Method of reading and counting with the follow-up having the count 0 delayed, as it must be during the early endgame:

Click Here To Show Diagram Code

[go]$$B start in the environment, -23
$$ +-------------------------+
$$ | O C B B C C C 2 X . . X |
$$ | O O C O B C O C O X X X |
$$ | . O O O C O O B O X . X |
$$ | . O X O O O C B O X . X |
$$ | . O X X X O B B O X . X |
$$ | . O X X 1 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]$$B start locally with the gote option, -23
$$ +-------------------------+
$$ | O C B B C C 2 3 X . C X |
$$ | O O C O B C O 1 O X X X |
$$ | . O O O C O O X O X . X |
$$ | . O B O O O . X O X . X |
$$ | . O B B B O X X O X . X |
$$ | . O B B 4 O O O O X . X |
$$ | O O O O X X X X X X X X |
$$ +-------------------------+[/go]

3 EDITs: 2x early endgame; minor corrections.

Seeing the wordings MGOTE, MGOTE_GOTE_OPTION, MGOTE_SENTE_OPTION it seems a good place here, in this thread dealing with gote and sente moves to define clearly what is meant by this notion.

[img][/img]

Let's take this basic tree in the attachement, in which you can see a black gote option AB and another black option AC, and similarly a white gote option AE and another white option AD.
OC, at the beginning of the analysis you do not know if options AC and AD are gote or sente;

Let's call a, b, c ... the counts of positions A, B, C ...
You can calculate immediately c = (f+g)/2 and d = (h+i)/2

Now how do you proceed in order to know if AC is gote or sente? I understood you will use a MGOTE_SENTE_OPTION value and a MSENTE value but how will you calculate these values? I guess you will use c and g values but what about d and e, especially when you do not know yet if AD is sente or gote?

To calculate values of a player's option, temporarily prune the tree by removing the currently not considered alternative option. This leaves two options for the opponent's start.

Your tree has options for both starting players and this makes it more complicated than any of my theory uses explicitly. Nevertheless, your questions are valid for a possibly broader understanding.

With two options for the opponent's start remaining in the pruned tree, CGT techniques must be considered. In particular, we try to detect if any of the opponent's options is dominated or reversible so can also be pruned.

If thereby the tree is without alternative options, calculate the tentative gote move value, Black's tentative sente move value (for the left sente sequence) and White's tentative sente move value (for the right sente sequence) like we always calculate gote or sente move values, respectively.

If, however, two options of the opponent remain and other reasoning cannot choose clearly, then tentative move values are undefined (so far).

If all tentative move values, tentative counts and Black's and White's follow-up move values could be calculated for a particular option, we derive the assessments of the type of the initial position for that option.

"Definitions 17 [types]
For such a local endgame, we define these types:
local gote :<=> MGOTE < MB,SENTE, MW,SENTE,
Black's local sente :<=> MW,SENTE ≥ MGOTE > MB,SENTE,
White's local sente :<=> MB,SENTE ≥ MGOTE > MW,SENTE,
Black's ambiguous :<=> MW,SENTE > MGOTE = MB,SENTE,
White's ambiguous :<=> MB,SENTE > MGOTE = MW,SENTE,
doubly ambiguous :<=> MGOTE = MB,SENTE = MW,SENTE." [22]

Again, I do NOT define the types of individual moves. I define the types of positions.

https://www.lifein19x19.com/viewtopic.p ... 45#p143245

[img][/img]
OK lets'take the following example: b = +3, e = -1, f = +28, g = h = +2, i = -24
What are the values of MGOTE, MB,SENTE and MW,SENTE ?