OK. We can write G this way, using more slashes.

G = {21 ||| 18 | 4 || 0 | -14}

I'll draw the thermograph. Be back soon.

I'm back. Here is the thermograph.



I think I need to add the thermograph for {18|4||0|-14}, as well. Back soon.



That's more like it. OC, this is a gote position, but it has a double sente temperature range from 0 to 7.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

M would be the correct move value. Initially, we do not necessarily know if we have a local gote with a gote move value Mgote, Black's local sente with Black's sente move value Mb,sente or White's local sente with White's sente move value Mw,sente. Therefore, we start with tentative move values, or one such compared to the follow-up move values.

I am not fluent with thermographs, let Bill do that.

You write G = {21|{{18|4} | {0|-14}} but I think you mean

{21} + {18|4||0|-14}.

Let me study the latter summand.

Fb = (18-4)/2 = 7.
Fw = (0-(-14))/2 = 7.

We have these intermediate counts:

Cb = (18 + 4) / 2 = 11.
Cw = (0 + (-14)) / 2 = -7.

Tentative Mgote = (Cb - Cw)/2 = (11 - (-7)) / 2 = 18/2 = 9.

Let us verify this:

Mgote > Fb, Fw <=> 9 > 7, 7 is fulfilled. Therefore, the initial local endgame is a local gote with
M := Mgote = 9.

To fulfil our curiosity, let us also study

Tentative Mb,sente = 4 - Cw = 4 - (-7) = 11.

Mb,sente < Fb <=> 11 < 7 is violated so the initial local endgame is not Black's local sente.

And let us study:

Tentative Mw,sente = Cb - 0 = 11 - 0 = 11.

Mw,sente < Fw <=> 11 < 7 is violated so the initial local endgame is not White's local sente.

G = {21|{{18|4} | {0|-14}} and {21} + {18|4||0|-14} seems quite different.
As Bill wrote G = {21|{{18|4} | {0|-14}} can be written G = {21 ||| 18 | 4 || 0 | -14} if you prefer.

OK Bill, now take the complete game (see the game I proposed but here with a simplier upper side)
{21 ||| 18 | 4 || 0 | -14} + {20|0} + {0|-6} + {4|0}
What is the best sequence for this game if it is white to play?

Yes, the two games / ensembles of games are quite different indeed. Your diagram position contained a separate large simple gote endgame and a separate want-be double sente region. Separate games are combined by forming their sum.

G = {21 ||| 18 | 4 || 0 | -14} can be studied for different purposes but, with another branch of the tree, do not fall into the scope of my non-existence proofs. Anyway, we can also study values of that larger game tree.

So let me calculate the count of the right part {18 | 4 || 0 | -14}, of which I have already determined that it is a local gote:

C = (Cb + Cw)/2 = (11 + (-7)) / 2 = 2.

Now, we can proceed studying G = {21 ||| 18 | 4 || 0 | -14}.

It might be White's long gote, else White's local sente, else a simple gote.


We already know

21 count after move 1 by Black of Black's alternating sequence
2 count after move 1 by White of White's alternating sequence
11 count after move 2 by Black of White's alternating sequence
4 count after move 3 by White of White's alternating sequence

Gw2 = 11 - 2 = 9 gain of move 2 by Black of White's alternating sequence
Gw3 = 11 - 4 = 7 gain of move 3 by White of White's alternating sequence


First, we make the hypothesis of White's long gote.

M(G) = (21 - 4) / 2 = 8 1/2 tentative gote move value
C(G) = (21 + 4) / 2 = 12 1/2 tentative gote count
Gb1 = 21 - 12 1/2 = 8 1/2 tentative gain of move 1 by Black of Black's alternating sequence
Gw1 = 12 1/2 - 2 = 10 1/2 tentative gain of move 1 by White of White's alternating sequence

M(G) <= Gb1, Gw1, Gw2, Gw3 <=> 8 1/2 <= 8 1/2, 10 1/2, 9, 7 is partially violated so we do not have White's long gote.


Second, we make the hypothesis of White's local sente.

M(G) = 21 - 11 = 10 tentative sente move value
C(G) = 11 tentative sente count
Gb1 = 21 - 11 = 10 tentative gain of move 1 by Black of Black's alternating sequence
Gw1 = 11 - 2 = 9 tentative gain of move 1 by White of White's alternating sequence

M(G) <= Gb1, Gw1, Gw2 <=> 10 <= 10, 9, 9 is partially violated so we do not have White's local sente.


Third, we make the hypothesis of a simple gote.

M(G) = (21 - 2) / 2 = 9 1/2 tentative gote move value
C(G) = (21 + 2) / 2 = 11 1/2 tentative gote count
Gb1 = 21 - 11 1/2 = 9 1/2 tentative gain of move 1 by Black of Black's alternating sequence
Gw1 = 11 1/2 - 2 = 9 1/2 tentative gain of move 1 by White of White's alternating sequence

M(G) <= Gb1, Gw1 <=> 9 1/2 <= 9 1/2, 9 1/2 is (of course) fulfilled so G = {21 ||| 18 | 4 || 0 | -14} is a simple gote with M(G) = 9 1/2 as the move value and C(G) = 11 1/2 as the count.

Seems to agree to Bill's thermograph:)

Surely you know where I encountered difficulties Bill

Starting from G = {21 ||| 18 | 4 || 0 | -14}
if use tax = 7 then the game becomes
G = {14 ||| 11 | 11 || 7 | 7}

Now what will happen with tax = 8?
I see that the games {11 | 11} and {7 | 7} are blocked and I do not know how to go to tax = 8.
Can you help me Bill?

Let's call the games, G, H, M, and N. I'll us a minus sign to indicate a White moves, a plus sign to indicate a Black move. We start with a mean territorial value of 11½ + 10 - 3 + 2 = 20½.

First, let White play the ever popular hotstrat strategy, taking the average gains in order.

Edit: Oops. Correction:

1) - H + G - M + N = 20½ - 10 + 9½ - 3 + 2 = 19

Can White do better? Let's see.

2) - G + H - G + G - M + N = 20½ - 9½ + 10 - 9 + 7 - 3 + 2 = 18

If I have added and subtracted right and used the right gains and losses, that's 1 point better.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

OK Robert now you can reach my point:
in the game {21 ||| 18 | 4 || 0 | -14} + {20|0}
where will you play first? In the simple 9½ gote G={21 ||| 18 | 4 || 0 | -14} or in the simple 10 gote {20|0} ?

Sure. Replace {11|11} and {7|7} with their mean (mast) values. That corresponds with using the mast instead of the lower walls. That yields.

G = {14 || 11 | 7}

When t = 8 we have

G = {13 || 11 | 9}

When t = 9 we have

G = {12 || 11 | 11} -> {12 | 11}

When t = 9½ we have

G = {11½ | 11½}

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Black starts:

B1) 21 - 0 = 21.

B2) 20 + {18 | 4 || 0 | -14} -> 20 + {18 | 4} -> 20 + 4 = 24. Correct.

White starts:

W1) 0 + 21 = 21.

W2) {18 | 4 || 0 | -14} + 20 -> {0 | -14} + 20 -> 0 + 20 = 20.

W3) {18 | 4} + {20|0} -> 0 + 18 = 18. Correct.

Yes Bill here is my point. In this example you have to begin by this 9½ gote instead of the simple 10 gote.
This is very strange. Isn't it?
It seems to me it is due to the potential double sente hidden in the game G.

take the optimal sequence : - G + H - G + G - M + N
At the beginning this double sente is sleeping because the temperature (t=10) is too high.
After the first move (=> -G) the temperature is still high and the double sente is still sleeping
But after the second move (+H) in an other area (!) the temperature drops and the double sente wakes up suddenly!

Oops, too easy isn'it?
Thank you Bill. It is very clear.

It is well known that decreasing order of move values is not always the best. Anomalies already occur for one local endgame with with one player's follow-up in an ideal environment!

Well, first, I goofed and used 11½ (the mean value) instead of 9½ (the gain) in my calculations. I realized that when I looked over my note and went back and corrected it.

But for a play with a slightly smaller gain than another play to be preferred to that play is not strange. It is not usual, but it is not too uncommon.



Edit:

The double sente aspect has nothing to do with it. For instance, let's replace {18|4||0|-14} with {18|4||-7}. That makes G = {21|||18|4||-7}. Remember, H = {20|0}, M = {0|-6}, and N = {4|0}.

First White plays the hotstrat strategy of playing the largest play (on average) first.

1) - H + G - M + N = 20½ - 10 + 9½ - 3 + 2 = 19

Can White do better? Let's see.

2) - G + H - G + M - N = 20½ - 9½ + 10 - 9 + 3 - 2 = 13

That's 6 points better, not just 1 point better.

Edit2: Hmmm. Can Black do better than 2)?

3) - G + G - H + G - M + N = 20½ - 9½ + 9 - 10 + 7 - 3 + 2 = 16

Yes, indeed. That's only 3 points worse.

Interestingly, best play at temperature 0 involves playing the next to largest play twice, once by White and once by Black.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Yes Robert I understand but at least try to be more explicit. When you claim "double sente" does not exist without explaining in what context it is true it becomes really ununderstandble. It is only with that last post I understood your context for such proof, context which is quite small comparing to the real areas we can find in a real game.

Very good Bill I agree.
At least, have I managed to show how a double sente (with our definition) move may appear when temperature drops?
The difficulty is then : which side will made the temperature drop under the value which wakes up the double sente move (bad luck)? Not easy indeed.

The context of each proof is the proposition and its presuppositions, possibly together with applied earlier axioms, definitions and propositions. The context is the task of assigning the correct count, move value, gains and type to a local endgame (of a certain class). Now that you have possibly understood it, what do you think of my proof? Is it - for its declared scope - correct? Can you appreciate the elegant constructions of proofs by contradiction? The case "b = w" is the most difficult and beautiful part. Bill, did you want me to rediscover its beauty or have you just overlooked it?

Global double sente or what definition do you refer to?

Always the same Robert I try to do my best not changing the definition:

we can characterise a local double sente endgame be these value conditions: M < Fb, Fw

Well, I found a better play for Black. See edit.

BTW, from your examples I get the impression that you are very comfortable with the idea of the sum (combination) of games. For those the concepts of global gote, global sente, and global double sente can be helpful. But, OC, they depend on the global situation.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.