Black can do better at temperature 0.

Oops yes Bill, I am a stupid boy!
(-6)in sente and then (+10) => +4

What about this combination?

{+22|-6},-3|-7} + {10|0}

Now how about integer temperatures up to 5?

The definition [of a double sente] should be as near as possible as the common understanding of go players

Since not all players use values but players with a weak understanding of endgame only use an informal understanding, the common go players' understanding of double sente would be informal. However, some players have not reflected yet that local versus global considerations of double sente differ. Therefore, the common go players' understanding of double sente does not exist. Concerning global considerations, some players are aware that one should not always play a double sente immediately because it might be relatively small while other players (with a weak understanding of endgame) are not aware of that and instead believe overly simplistic traditional advice to play in double sente as early as possible. Only for local considerations, we can identify some common go players' understanding of double sente: that either player's local play is sente meaning an immediate reply by the opponent. In only informal terms, we cannot better characterise why an immediate local reply should be necessary.

In terms of values, we can characterise why an immediate local reply should be necessary: after either player's local play, the reply is more valuable. That is, the move value in the initial local endgame position is smaller than both replies' follow-up move values. Let us use these variables:

M := the move value in the initial local endgame position.
Fb := the move value in the follow-up position created after Black's start.
Fw := the move value in the follow-up position created after White's start.

Now, we can characterise a local double sente endgame be these value conditions:

M < Fb, Fw.

(This annotation abbreviates "M < Fb and M < Fw".)

However, simply speaking, the mathematically proven theorem says:

A local double sente endgame with M < Fb, Fw does not exist.

The common go players' understanding did not know this yet:)

OK. Thermographs do two things. The thermograph of a game finds the mean value of the game, and it finds the result of minimax play in the game, starting with each player, at each temperature.

What else does your skeptic desire?

As soon as you decide to define what is a double sente it exists doesn'it?

Click Here To Show Diagram Code

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

The game above looks like G1 = {{+22|-6},-3|-7}
And my goal is to compare this game G1 with the game G2 = {-3|-7}

I do not feel it is skepticism Bill. Maybe it is only that I expected more from the theory ?

To do so, you subtract G2 from G1. I. e., you consider the sum,

H = {{22|-6},-3|-7} + {7|3}

If White (Right) plays first, White cannot win. That is, if White plays to -7 in the game on the left, Black (Left) replies to 7 in the other game, for jigo (0). If White plays to 3 in the game on the right, Black replies to -3 in the other game, also for jigo.

If Black plays first she wins. She plays to {22|-6} + {7|3}. White's best play is to -6 on the left, after that Black plays to 7 in the other game, for a score of 7 - 6 = 1.

Therefore G1 > G2.

This question is answered with a difference game, not with thermography.

That is exactly what I tried to explained in my previous posts.
Though G1 > G2 the thermography claims meanValue(G1) = meanValue(G2).

When you look at the theory as a whole (CGT theorems, thermography, difference game etc) this result might appear as a contradiction.

I just say it is a pity result that may logically cause some skepticism for the non mathematician go players who see in other cases a very acurate calculation of mean values and I fear it is difficult to convinced a non mathematician that it is not necessarily a contradiction (I know the assumptions behind thermography and difference game are quite different OC)

Unlike proofs by counter-example, the theorem cannot be proven by example because it applies to all such examples. Instead, the proof is by abstract verification. More later.

Can you please simplify your example so that there are not many follow-up moves and safely alive surrounding strings?

I must be wrong somewhere but where Robert?

I attach my mathematical proof of non-existence of local double sente, which proceeds Bill's preliminary study.

Now the temperature has dropped so that the double sente move is of greatest importance and white is quite lucky to be able to play this double sente move of course before playing in the remaining gote!

If mathematics prove this double sente cannot exist I must be wrong somewhere but where Robert?

Here under I tried to build such example:

Click Here To Show Diagram Code

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

The idea is the following

Click Here To Show Diagram Code

[go]$$W White to play
$$ ------------------------
$$ | . . . . . . 3 . . . .|
$$ | . . . . . . X O . . .|
$$ | X X . . . . X O X . .|
$$ | . X X X X X X 1 X . .|
$$ | X X O O X X O O O O O|
$$ | X O O O X X . X X O .|
$$ | X O O O O 2 O O X O O|
$$ | X O X X X X X O O O O|
$$ | X X X X X X X X X O O|
$$ | X X X X X X X X X O O|
$$ | . X X X X X O O . O .|
$$ ------------------------[/go]

white is the biggest gote point and create the double sente (the two hane-tsugi)
The temperature is still too high (t=10) and unfortunetly black has to play first
Now the temperature has dropped so that the double sente move is of greatest importance and white is quite lucky to be able to play this double sente move of course before playing in the remaining gote!

If mathematics prove this double sente cannot exist I must be wrong somewhere but where Robert?

Do you mean precedes?

Since not all players use values but players with a weak understanding of endgame only use an informal understanding, the common go players' understanding of double sente would be informal. However, some players have not reflected yet that local versus global considerations of double sente differ. Therefore, the common go players' understanding of double sente does not exist. Concerning global considerations, some players are aware that one should not always play a double sente immediately because it might be relatively small while other players (with a weak understanding of endgame) are not aware of that and instead believe overly simplistic traditional advice to play in double sente as early as possible. Only for local considerations, we can identify some common go players' understanding of double sente: that either player's local play is sente meaning an immediate reply by the opponent. In only informal terms, we cannot better characterise why an immediate local reply should be necessary.

In terms of values, we can characterise why an immediate local reply should be necessary: after either player's local play, the reply is more valuable. That is, the move value in the initial local endgame position is smaller than both replies' follow-up move values. Let us use these variables:

M := the move value in the initial local endgame position.
Fb := the move value in the follow-up position created after Black's start.
Fw := the move value in the follow-up position created after White's start.

Now, we can characterise a local double sente endgame be these value conditions:

M < Fb, Fw.

(This annotation abbreviates "M < Fb and M < Fw".)

However, simply speaking, the mathematically proven theorem says:

A local double sente endgame with M < Fb, Fw does not exist.

The common go players' understanding did not know this yet:)