Uh, no, I do not claim that the correct result is mine but I bow to thermography. My result is just "consistent within my method for my definitions of values".

My method is the best one for practical application if faster / simpler methods are inapplicable. E.g., for simple local endgames, making a hypothesis works but is overkill; a simple gote / sente comparison will do.

Your method is not plainly wrong but you advertise using a (somewhat) rich environment without actually ever using this fact. You must prove your point by using your assumptions. In particular, you must explain why a "sufficiently" rich environment is possible to use instead of a rich, arbitrarily dense environment of thermography.

I do not exclude that simpler ideas exist:
- simple local endgame evaluation
- method of comparing the opponent's branches
- my pragmatic method of comparing counts
- my pragmatic method of comparing move values
- CGT reversal (simpler in the sense of simplifying the tree) by difference game

Whether your alleged method is simpler I do not know. First perform it well, then we can judge better. However, how can some (rather) rich environment ever be simpler than no environment?

I prefer to avoid the more powerful rich environment because play in them by means of proving theorems or constructing thermographs algebraically or graphically is hard. Each time, I have to look up how it works. A have done it for a few examples and proofs of theorems but just because I can do it with great effort does not mean that I would like it or even pretend simplicity of execution. In the sample for [22], you see a bit of my related "exercises", for which I have tried to keep the theory of thermography as simple as possible but it remains complicated with linear algebra at each calculation step.

I see here some misunderstanding. OC it is not surprising because I am still in the building process of my method.
First of all I believe that my “sufficiently rich environment” is only another formal definition of the rich, arbitrarily dense environment. Anybody can see what a rich environment is and everybody can build a rich environment as rich as she wants. But, for a non-mathematician, going to the limit to reach an arbitrarily dense environment is not that obvious. Basically there are no difference.
It is true that my "move value" definition uses explicitly a “sufficiently rich environment”, but I also said that the “move value” itself depends only on the subtree of the position.
IOW I can build my “move value” using a calculation ignoring the environment and then I have only to verify that the result fulfilled my definition of “move value”.
BTW I never found a position whose miai value (calculated by thermography) does not fulfilled my famous property 1. Maybe it is obvious because I said my “sufficiently rich environment” is basically identical to the rich, arbitrarily dense environment of thermography. That means that I can keep my definition of move value as it stands (with “sufficiently rich environment”) and choose thermography as calculation method, waiting confidently for a counter example.
Why then I work on another method? Because thermography cannot be used in practice. The reason for that is that thermography is very ambitious, being able to give not only the “miai value” and the count but also information at whatever temperature.
In order to build a far simpler method my goal is only to calculate the move value (= miai value) and the count, and nothing else.

" I can build my “move value” using a calculation ignoring the environment and then I have only to verify that the result fulfilled my definition of “move value”."

Good aim. Please work out your method, especially how to verify using the rich environment!

"In order to build a far simpler method my goal is only to calculate the move value (= miai value) and the count, and nothing else."

I see, but calculating gains may also be useful.

You may also wonder why I defined a move value (see https://lifein19x19.com/viewtopic.php?p=277735#p277735) instead of using the already known miai value. Here again it's a matter of simplicity for a go player.
I expect that a rich environment at a given temperature is easy to undertand, and I expect that to have to play locally under a certain temperature called "move value" is also easy to understand. As a consequence a go player may easily imagine that, facing a position P1 and a position P2, if the move value of P1 is greater than the move value of P2, then it will quite often happen that a move in P1 will appear before a move in P2 to reach the best score.
In the other hand, explaining what a "miai value" is, needs far more than few lines and in addition the goal looks not that obvious for a go player (how a non theorician go player can really understand what the base of a mast means?)

What is the "move value" of a simple ko position A


let's call c and d the counts of leaves C and D.
How to justify that the "move value" va of position A is equal to the value m = (d - c) / 3 ?

1) let's take first a temperature t > m
Let's take a rich environment Rich(t, N)
In this environment the score of the game if black decides to gain the ko is
score1 = d - g(Nt/N) - g((N-1)t/N) + g((N-2)t/N) - ...
while the score of the game when black plays in the environment and white finishes the ko is
score 2 = c + g(Nt/N) + g((N-1)t/N) - g((N-2)t/N) - ...
=> score 2 - score1 = c - d + 2(Nt/N) + 2((N-1)t/N) - 2((N-2)t/N) + …
=> score 2 - score1 = c - d + 2t + 2t/N ((N-1) - (N-2) + (N-3) – (N-4) …)
If N is odd then ((N-1) - (N-2) + (N-3) – (N-4) …) = (N-1)/2
And if N is even then ((N-1) - (N-2) + (N-3) – (N-4) …) = N/2
in any case we have ((N-1) - (N-2) + (N-3) – (N-4) …) ≤ N/2
=> score 2 - score1 ≤ c - d +2t + 2t/N (N/2) = c - d + 3t < c – d + 3m = 0
Eventually t > m => score2 > score1 => black plays in the environment to reach the best score.
2) With the same kind of reasoning you can also prove that
t < m => black must play in the ko to reach the best score
3) and 4) same result with white to play
Finally the “move value” of position A is va = m = (d - c) / 3
OC such proof is quite boring but I am very convinced it works.

Currently, I lack time to check your proof draft carefully but I get your badic construction. One thing I notice immediately that I do not understand.

You presume t>m to imply c - d + 3t < c – d + 3m. How?

Oops you are right Robert. Here is a correction:
Let's take a rich environment Rich(t, N)
In this environment the score of the game if black decides to gain the ko is
score1 = d - g(Nt/N) - g((N-1)t/N) + g((N-2)t/N) - ...
while the score of the game when black plays in the environment and white finishes the ko is
score 2 = c + g(Nt/N) + g((N-1)t/N) - g((N-2)t/N) - ...
=> score 2 - score1 = c - d + 2(Nt/N) + 2((N-1)t/N) - 2((N-2)t/N) + …
=> score 2 - score1 = c - d + 2t + 2t/N ((N-1) - (N-2) + (N-3) – (N-4) …)
If N is odd then ((N-1) - (N-2) + (N-3) – (N-4) …) = (N-1)/2
And if N is even then ((N-1) - (N-2) + (N-3) – (N-4) …) = N/2
in any case we have ((N-1) - (N-2) + (N-3) – (N-4) …) >= (N-1)/2
=> score 2 - score1 ≥ c - d + 2t + (2t/N)(N-1)/2) = c - d + 2t + t(N-1)/N
=> score 2 - score1 ≥ c - d + 3t – t/N
=> score 2 - score1 ≥ 3t – 3m – t/N
Choosing N1 > t / (3t -3m)
N > N1 => score 2 - score1 > 0
That way we can clearly see the consquence of being able to choose an environment as rich as you want.
Thank you Robert for your help.

Following the link given by jlt in the post https://lifein19x19.com/viewtopic.php?p=277423#p277423 you can find the following position

Click Here To Show Diagram Code

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

and the move value given is 20 points (deiri value).

My own analysis gives only 18 points and so I must be wrong somewhere.
What is your own result?

I have now made great progress in exploring my method to calculate move values and I am testing this method on various positions.
Taking the flashcards on the link gomagic-endgame-flashcards-lite-pack.zip I encounter some issues.
Here is one of them (number 24)

Click Here To Show Diagram Code

[go]$$
$$ . . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X . .|
$$ . . O X O . .|
$$ . . . . O . .|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|
$$[/go]

The site gives as answer 8 points gote (deiri value) where my result is 5 points in reverse sente <=> 10 points gote (deiri value).
What is your view?

Here are my calculations. No guarantee that they're correct, but they do match the flashcard.

Click Here To Show Diagram Code

[go]$$B B+3 (the marked stones are Black's privilege)
$$ . . . . . . .|
$$ . . . . . . .|
$$ . . . . X T T|
$$ . . . . X T T|
$$ . . O O X 3 T|
$$ . . O X O 1 #|
$$ . . . . O 2 #|
$$ . . . O . @ @|
$$ . . . . . T T|
$$ . . . . . . .|
$$[/go]

Click Here To Show Diagram Code

[go]$$W W+5 (the marked stones are an even split)
$$ . . . . . . .|
$$ . . . . . . .|
$$ . . . . X T T|
$$ . . . . X 2 #|
$$ . . O O X 1 @|
$$ . . O X O 3 T|
$$ . . . . O T T|
$$ . . . O . T T|
$$ . . . . . T T|
$$ . . . . . . .|
$$[/go]

That gives a total swing of 8 points in gote.

That was exactly my first view but when you consider white to move it seems to me better to play:

Click Here To Show Diagram Code

[go]$$W W+2 (the marked stones are an even split)
$$ . . . . . . .|
$$ . . . . . . .|
$$ . . . . X T T|
$$ . . . . X T T|
$$ . . O O X 2 #|
$$ . . O X O 1 @|
$$ . . . . O T T|
$$ . . . O . T T|
$$ . . . . . T T|
$$ . . . . . . .|
$$[/go]

That way white gets 5 points in sente (<=> 10 points gote)

Sente must be verified or rejected.

White do you mean Robert? A flashcard is only made to train you to calculate a move value in a PRACTICAL game. I agree with you that verification would be appreciate in a theoritical approach but I guess such theoritical approach would need far too much time.
IOW, assuming you are white in a practical game:

Click Here To Show Diagram Code

[go]$$ . . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X a .|
$$ . . O X O b .|
$$ . . . . O . .|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|
$$[/go]

do you prefer a or b (assuming some kind of ideal environment)?

I prefer the naive assumption that, if it is a local sente, it must be Black's...! White's region is wider open so Black should have the larger follow-up so any local sente would be Black's. Flashcard or not - the basics must be right! I.e., if a flashcard merely says 5 pt in [local] sente, the card creator must still verify a) it is a local sente, b) whether it is either Black's or White's local sente and c) write the right sente player on the flashcard.

Click Here To Show Diagram Code

[go]. . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X a .|
$$ . . O X O b .|
$$ . . . . O . .|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|[/go]

My initial question was : "as WHITE, do you prefer to play at "a" or do you prefer to play at "b"? (as far as I am concerned I prefer white "b")

It is not that easy to understand your answer. I agree with you that a black move at "a" would be "more" sente that a white move at "b" but how can you justify your conclusion : "if it is a local sente, it must be Black's...!"
This could be true if the best black move would be the sente move black "a" but you never claimed that the sente black move at "a" was better than the gote black move at "b" did you?
In any case and in order to make some progress, can you please clarify your answer to my initial question?

For local evaluation, one must always consider both Black's and White's start, like one does in CGT.

In a global context, one can consider one particular player's start and whether his usually correct local gote / ambiguous / sente play is also correct in the particular global context.

For the local verification of the options with follow-up as a local gote / ambiguous / local sente, for example, compare the tentative gote move value to Black's follow-up move value and White's follow-up move value. Here, we also have other first move options simply speaking without follow-up because they are (3-move) gote options. We also need to compare, if indeed, sente option to the gote option and the gote-sente-difference in points.

Thereby you identify the correct type and values. Except that you don't because you forgot Bill's theory and do not look up mine;) It depends on whether we are in the early or late endgame.

You may think that I do not agree with you but it is not true. Seeing your approach is purely theoritical (you very often remind the work made in CGT, Bill's theory or yours) I completly agree with you that a lot of work has to be done in order to verify and confirm if a black or white sente move is better than a black or white gote move, or if other sequences of moves have to be taken into account etc...
But here is the misunderstanding: providing a flashcard is not obvious the purely theoritical approach is clearly impossible.

As an example let's consider flashcard number 33:

Click Here To Show Diagram Code

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

Can CGT, Bill's theory or your theory can help us to find the move value of such position? Surely not because it is far too difficult. I guess even a hundred pages of analyses will not be enough to calculate the correct move value.

Click Here To Show Diagram Code

[go]. . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X a .|
$$ . . O X O b .|
$$ . . . . O . .|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|[/go]

this position is simplier but I guess that dozens of pages would be necessary to give the correct move value. Even without finding the correct move value it is already difficult to say if white move "a" is better than white move "b" and I clearly understand why you did not answer my question : the answer is simply too difficult to find in a pure theoritical approach.
BTW which go player will accept to read in detail such dozens pages of analysis?

As a consequence some other approaches are needed and discussions can take place to improve such approaches and to improve the first evaluation found.

But if you contest a result by saying it is not correct for a theoritical point of view then, though you are right, it is only a very negative approach because this theoritical approach is completly impractical and cannot bring any help.

Flashcards are for go players and not for theoriticians.
I am pretty sure you can bring nice ideas on some flashcards but as a go player not as a theoritician.

In your previous message, your first diagram is more complicated than the second. I do not think that both are as complicated to need "dozens of pages". OTOH, they are also not so simple, from a POV of rather accurate evaluation, to allow for a naive "flashcard" guess. However, they are complicated enough to require somewhere between minutes and hours of thinking, and it is hard to predict how much time will be needed. As I have explained before, it takes only seconds to show a problem diagram but I do not have the time to solve each problem accurately whenever a new problem is shown. It is similar to intermediate to advanced life and death problems: not everybody has the time to solve all of them and explain the solution in detail just because the problems are shown.

However, one can make simplifying assumptions:
- Suppose we know which follow-ups are relevant and study only them.
- Suppose the options with follow-ups are sente for one starting player (here: Black) and gote for the opponent.

The following assumption is risky:
- For each starting player, suppose that one of the two options dominates and we know which it is.

Let's me show you how I analysed this position with my method (I am now not very far for being able to give you an article on the subject).

Click Here To Show Diagram Code

[go]. . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X . .|
$$ . . O X O . .|
$$ . . . . O . .|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|[/go]

First of all my analyse is based on area scoring (and not territory scoring)

At first sight I was convinced that the best black move as well as the best white move where the hane gote moves showed by dfan in https://lifein19x19.com/viewtopic.php?p=278108#p278108
The result in area approach is the following:

Click Here To Show Diagram Code

[go]B . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X M M|
$$ . . O X O M M|
$$ . . . . O . M|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|[/go]

The five marked intersections represent the intersections each player can gain by playing first.
On average (I mean the count of the position) 2.5 of these intersections are for black and 2.5 are for white.

Click Here To Show Diagram Code

[go]B . . . . . .|
$$ . . . . . . .|
$$ . . . . X . .|
$$ . . . . X . .|
$$ . . O O X a .|
$$ . . O X O b .|
$$ . . . . O . .|
$$ . . . O . . .|
$$ . . . . . . .|
$$ . . . . . . .|[/go]

What about the black sente move at "a" or the white sente move at "b" ?
If black plays at the sente point "a" and white answers at "b" then, comparing to the previous diagram black takes only two of the five marked intersections => count is better for white => black "a" is wrong
On the other hand if white plays at the sente point "b" and black answers at "a" then white takes three of the five marked intersections => count is better for white => white "b" is correct providing it is sente. Obviously the potential sente ogeima after white "b" proves that white "b" is sente => white "b" is probably the best move.
=> the position is sente for white

With rough analysis, you can prove everything incl. that the universe does not exist:)